Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

The multiplicative Zagreb indices of graphs with given connectivity or number of pendant vertices

The multiplicative Zagreb indices of graphs with given connectivity or number of pendant vertices For a graph G, the first multiplicative Zagreb index (G) is the product of squares of vertex degrees, and the second multiplicative Zagreb index (G) is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal Π (G) and Π (G) in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most k and p pendant vertices. In addition, Q Q the maximum and minimum values of (G) and (G) are provided. Our results extend and 1 2 enrich some known conclusions. Keywords: Connectivity; Edge connectivity; Extremal bounds; Multiplicative Zagreb indices; Pendant vertices. AMS subject classification: 05C35, 05C38, 05C75, 05C76, 05C09, 05C92 1 Introduction A topological index is a single number which can be used to describe some properties of a molecular graph that is a finite simple graph, representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. In recent decades, these numerical quantities have been found useful for the study of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) and for the structural essence of biological and chemical compounds. The well-known Randi´c index is one of the most important topological indices. In 1975, Randi´c introduced a moleculor quantity of branching index [1], which has been known as the famous Randi´c connectivity index and that is a most useful structural descriptor in QSPR and QSAR, see [2, 3, 4, 5]. Mathematicians have considerable interests in the structural and applied aspects of Randi´c connectivity index, see [6, 7, 8, 9]. Based on the successful considerations, Zagreb indices [10] are introduced as an expected formula for the total π-electron energy of conjugated Corresponding author. Email addresses: [email protected](S. Ji), [email protected](S. Wang), [email protected](T. Muche), [email protected](S. Hayat). arXiv:1711.09014v2 [math.CO] 6 Aug 2021 molecules as follows. X X M (G) = d(u) and M (G) = d(u)d(v), 1 2 u∈V (G) uv∈E(G) where G is a (molecular) graph, uv is a bond between two atoms u and v, and d(u) (resp. d(v)) is the number of atoms that are connected with u (resp. v). Zagreb indices have also been employed as molecular descriptors in QSPR and QSAR, see [11, 12]. Recently, Todeschini et al. (2010) [13, 14] proposed the following multiplicative variants of molecular structure descriptors: Y Y Y Y Y 2 d(u) (G) = d(u) and (G) = d(u)d(v) = d(u) . 1 2 u∈V (G) uv∈E(G) u∈V (G) In the interplay among mathemactics, chemistry and physics, it is not surprising that there are numerous studies of properties of the (multiplicative) Zagreb indices of molecular graphs [15, 16, 17, 18, 19, 20, 21, 22, 32, 34, 40]. In view of these results, researchers are interested in finding upper and lower bounds for multi- plicative Zagreb indices of graphs and characterizing the graphs in which the maximal and minimal index values are attained. In view of the above problems, various mathematical and computational properties of Zagreb indices have been investigated in [23, 24, 25]. Other directions of investigation include studies of relation between multiplicative Zagreb indices and the corresponding invariant of elements of the graph G (vertices, pendant vertices, diameter, maximum degree, girth, cut edge, cut vertex, connectivity, perfect matching). For instance, the first and second multiplicative Zagreb indices for a class of chemical den- drimers are explored by Iranmanesh et al. [26]. Considering trees, unicyclic graphs and bicyclic graphs, Borovi´canin et al. [27] introduced the bounds on Zagreb indices with a fixed domination number. The maximum and minimum Zagreb indices of trees with given number of vertices of maximum degree are proposed by Borovic´anin and Lampert [28]. Xu and Hua [29] introduced a unified approach to characterize maximal and minimal multiplicative Zagreb indices, respectively. Considering the trees of higher dimension, i.e. k-trees, Wang and Wei [30] provided the maximum and minimum values of these indices and the corresponding extremal graphs. Some sharp upper Q Q bounds for -index and -index in terms of graph parameters are investigated by Liu and Zhang 1 2 [31], including the order, size and radius of graphs. Ji and Wang [33] provided the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices. The bounds for the moments and the probability generating function of these indices in a randomly chosen molecular graph with tree structure of given order are studied by Kazemi [35]. Li and Zhao obtained sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number [36, 37]. In light of the information available for multiplicative Zagreb indices, and inspired by above results, in this paper we further investigate these indices of graphs with a given (edge) connec- tivity and number of pendant vertices. We give some basic properties of the first and the second Q Q multiplicative Zagreb indices. The maximum and minimum values of (G) and (G) of graphs 1 2 with given (edge) connectivity at most k and p pendant vertices are provided. In addition, the 2 corresponding extremal graphs are charaterized. In our exposition, we will use the terminology and notations of (chemical) graph theory (see [38, 39]). 2 Preliminaries Let G be a simple connected graph, denoted by G = (V (G), E(G)), in which V = V (G) is vertex set and E = E(G) is edge set. For a vertex v ∈ V (G), the neighborhood of v is the set N(v) = N (v) = {w ∈ V (G), vw ∈ E(G)}, and d (v) (or d(v)) is the degree of v with d (v) = |N(v)|. For G G G i ≥ 0, n denoted the number of vertices of degree i. For S ⊆ V (G) and F ⊆ E(G), we use G[S] for the subgraph of G induced by the vertex set S, G − S for the subgraph induced by V (G) − S and G − F for the subgraph of G obtained by deleting F. If G − S contains at least 2 components, then S is said to be a vertex cut set of G. Similarly, if G − F contains at least 2 components, then F is called an edge cut set. A graph G is said to be k-connected with k ≥ 1, if either G is complete graph K , or it has k+1 at least k + 2 vertices and contains no (k − 1)-vertex cut. The connectivity of G, denoted by κ(G), is defined as the maximal value of k for which a connected graph G is k-connected. Similarly, for k ≥ 1, a graph G is called k-edge-connected if it has at least two vertices and does not contain a (k − 1)-edge cut. The maximal value of k for which a connected graph G is k-edge-connected is said to be the edge connectivity of G, denoted by κ (G). According to the above definitions, the following proposition is obtained. Proposition 2.1 Let G be a graph with n vertices. Then (i) κ(G) ≤ κ (G) ≤ n − 1, (ii) κ(G) = κ (G) = n − 1 if and only if G K . k k Let V be the set of graphs with n vertices and κ(G) ≤ k ≤ n − 1. Denote E by the set of graphs n n with n vertices and κ (G) ≤ k ≤ n − 1. Note that if |V (G)| = n and |E(G)| = n − 1, then G is a tree. Let P and S be special trees: a path and a star of n vertices. The graph K is obtained by n n k k k joining k vertices of K to an isolated vertex, see Fig 1. Then K ∈ E ⊂ V . n−1 n n n n−k−j n−1 Figure.1 The graphs K and G(j, n − k − j) = K ⊕ H ⊕ K . j k n−k−j Q Q Considering the concepts of (G) and (G), the following proposition is routinely obtained. 1 2 3 k k Proposition 2.2 Let e be an edge of a graph G ∈ V (resp. E ). Then n n k k (i) G − e ∈ V (resp. E ), n n Q Q (ii) (G − e) < (G),i=1,2. i i In addition, by elementary calculations, these three statements are deduced. (x+m) Proposition 2.3 For m ≥ 0, F (x) = is monotonically increasing in (0, +∞). 1 x−1 (x−1+m) Proposition 2.4 If m ≥ 0, then F (x) = is a decreasing function in interval (0, +∞). 2 x+m (x+m) 2 2 n Proposition 2.5 F (x) = (x) (n − x) is monotonically increasing for x ∈ [1, ⌊ ⌋]. 3 Lemmas We first provide some lemmas, which are important in proving our main results. Q Q Lemma 3.1 [26] Let T be a tree on n vertices. If T is not P or S , then both (T ) > (S ) n n n 1 1 Q Q and (T ) > (P ) holds. 2 2 Q Q Considering the definitions of (G) and (G), we have the following lemma. 1 2 Lemma 3.2 Let u, v ∈ V (G) such that uv ∈/ E(G). Then Y Y Y Y (G + uv) > (G) and (G + uv) > (G). 1 1 2 2 Given two graphs G and G , if V (G ) ∩ V (G ) = φ, then the join graph G ⊕ G is a graph 1 2 1 2 1 2 with vertex set V (G ) ∪ V (G ) and edge set E(G ) ∪ E(G ) ∪ {uv, u ∈ V (G ), v ∈ V (G )}. 1 2 1 2 1 2 Lemma 3.3 Let G(j, n − k − j) = K ⊕ H ⊕ K be a graph with n vertices, in which K and j k n−k−j j n−k K are cliques, and H is a graph with k vertices, see Fig 2. If k ≥ 1 and 2 ≤ j ≤ , then n−k−j k Y Y (G(j, n − k − j)) < (G(1, n − k − 1)). 1 1 Proof. We consider the graph from G = G(j, n−k−j) to G = G(j−1, n−k−j +1). Note that if 1 2 v ∈ V (H ) in G , then d (v) = d (v); if v ∈ V (K ) in G , then d (v) = d (v) − 1 = j + k − 2; k 2 G G j 2 G G 2 1 2 1 Q Q if v ∈ V (K ) in G , then d (v) = d (v) + 1 = n − j. By the definitions of and , we n−k−j+1 2 G G 2 1 1 2 4 have Q Q Q 2 2 2 d(v) d(v) d(v) (G ) v∈V (K ) v∈V (H ) v∈V (K ) 1 j k n−k−j Q Q Q Q 2 2 2 (G ) d(v) d(v) d(v) 1 v∈V (K ) v∈V (H ) v∈V (K ) j−1 k n−k−j+1 j n−k−j 2 2 (j + k − 1) (n − j − 1) j−1 n−k−j+1 2 2 (j + k − 2) (n − j) j+(k−1) j−1 (j−1)+(k−1) =  . n−k−j+1 (n−j−k+1)+(k−1) n−k−j (n−j−k)+(k−1) n−k Since 2 ≤ j ≤ , we obtain j ≤ n − k − j < n − k − j + 1. By Proposition 2.3 and k ≥ 1, we have Q (G ) < 1, (G ) Q Q that is, (G ) < (G ). 1 2 1 1 We can recursively use this process from G to G , and obtain that 1 2 Y Y Y Y (G(j, n−k−j)) < (G(j−1, n−k−j+1)) < (G(j−2, n−k−j+2)) < · · · < (G(1, n−k−1)). 1 1 1 1 Q Q Therefore, (G(j, n − k − j)) < (G(1, n − k − 1)). Thus, we complete the proof. 1 1 Lemma 3.4 Let G be a connected graph and u, v ∈ V (G). Assume that v , v , . . . , v ∈ N(v)\N(u), 1 2 s 1 ≤ s ≤ d(v). Let G = G − {vv , vv , . . . , vv } + {uv , uv , . . . , uv }. If d(u) ≥ d(v) and u is not 1 2 s 1 2 s adjacent to v, then Y Y (G ) > (G). 2 2 Proof. By the concept of (G), we have d(u) Q d(u) d(u) d(v) d(u)+s (G) d(u) d(v) (d(u)+s) = = . d(v)−s d(u)+s d(v)−s (d(v)−s) (G ) (d(u) + s) (d(v) − s) d(v) d(v) By using d(u) ≥ d(v) > d(v) − s and Proposition 2.4, we obtain (G) < 1, (G ) Q Q which implies that (G ) > (G). This shows the lemma. 2 2 n−k Lemma 3.5 If k ≥ 1 and 2 ≤ j ≤ , we have Y Y (G(j, n − k − j)) < (G(1, n − k − 1)). 2 2 5 Proof. Let V (K ) = {v , v , · · · , v } and V (K ) = {u , u , · · · , u }. Note that vertex set j 1 2 j n−k−j 1 2 n−k−j {v , v , · · · v } ⊂ N(v )∩V (K ). We define a new graph G = G(j, n−k−j)−{v v , v v , . . . , v v }+ 2 3 j 1 j 1 2 1 3 1 j Q Q {u v , u v , . . . , u v }. By d(v ) ≤ d(u ) and Lemma 3.4, we have (G ) ≥ (G(j, n − k − j)). 1 2 1 3 1 j 1 1 2 2 ′ ′′ ′ Note that for G , v has neighbors in V (H ) only. Let G = G + {v u , 2 ≤ i ≤ j, 1 ≤ l ≤ 1 k i l Q Q Q ′ ′′ ′ n − k − j and v u ∈/ E(G )}. By Lemma 3.2, we have (G ) > (G ) ≥ (G). Therefore i l 2 2 2 Q Q ′′ G G(1, n − k − 1) and (G(j, n − k − j)) < (G(1, n − k − 1)). This completes the proof. 2 2 4 Extremal graphs with given connectivity In this section, the maximal and minimal multiplicative Zagreb indices of graphs with connectiv- k k ity at most k in V and E are determined, and the corresponding extremal graphs have been n n characterized in Theorems 4.1 and 4.5. Theorem 4.1 Let G be a graph in V . Then 2 2 2(n−k−1) (G) ≤ k (n − k) k(n − 2) and k k(n−1) (n−2)(n−k−1) (G) ≤ k (n − 1) (n − 2) , where the equalities hold if and only if G = K . Proof. Note that the degree sequence of K is k, n − 2, n − 2, · · · , n − 2, n − 1, n − 1, · · · , n − 1. | {z } | {z } n−k−1 k Q Q By the concepts of (G), (G) and routine calculations, we have 1 2 k 2 2k 2(n−k−1) (K ) = k (n − 1) (n − 2) and k k k(n−1) (n−2)(n−k−1) (K ) = k (n − 1) (n − 2) . Q Q Q Q k k It suffices to prove that (G) ≤ (K ) and (G) ≤ (K ), and the equalities hold if and n n 1 1 2 2 only if G K . n−1 ∼ ∼ If k ≥ n − 1, then G K K , and the theorem is true. If 1 ≤ k ≤ n − 2, then choose = = Q Q a graph G (resp. G ) in V such that (G ) (resp. (G )) is maximal. Since G ≇ K with 1 2 1 2 i n n 1 2 i = 1, 2, then G has a vertex cut set of size k. Let V = {v , v , · · · , v } be the cut vertex set of i i i1 i2 ik G . Denoted ω(G − V ) by the number of components of G − V . In order to prove our theorem, i i i i i we start with several claims. Claim 4.2 ω(G − V ) = 2 with i = 1, 2. i i 6 Proof. We proceed to prove it by a contradiction. Assume that ω(G − V ) ≥ 3 with i = 1, 2. Let i i G , G , · · · , G be the components of G − V . Since ω(G − V ) ≥ 3, then choose vertices 1 2 i i i i ω(G −V ) i i u ∈ V (G ) and v ∈ V (G ). Then V is still a k-vertex cut set of G + uv. By Lemma 3.2, we have 1 2 i i Q Q (G + uv) > (G ), a contradiction to the choice of G . Thus, this claim is proved. i i i i i Without loss of generality, suppose that G − V contains only two connected components, i i denoted by G and G . i1 i2 Claim 4.3 The induced graphs on V (G ) ∪ V and V (G ) ∪ V in G are complete subgraphs. i1 i i2 i i Proof. We use a contradiction to show it. Suppose that G [V (G ) ∪ V ] is not a complete subgraph i i1 i of G . Then there exists an edge uv ∈/ G [V (G ) ∪ V ]. Since G [V (G ) ∪ V ] + uv ∈ V , by i i i1 i i i1 i n,k Q Q Lemma 3.2, we have (G [V (G )∪V ]+uv) > (G [V (G )∪V ]), which is a contadiction. This i i1 i i i1 i i i shows the claim. By the above claims, we see that G and G are complete subgraph of G . Let G = K and i1 i2 i i1 n ′′ ′ ′′ G = K . Then we have G = K ⊕ G [V ] ⊕ K . i2 n i n i i n ′ ′′ Claim 4.4 Either n = 1 or n = 1. ′ ′′ ′ ′′ Proof. On the contrary, assume that n , n ≥ 2. Without loss of generility, n ≤ n . For (G), by Q Q ′ ′ Lemmas 3.3 and 3.5, we have a new graph G = K ⊕G [V ]⊕K such that (G ) > (G ) i 1 i i n−k−1 i i i i k ′ ′′ and G ∈ V . This is a contradition to the choice of G . Thus, either n = 1 or n = 1, and this i i claim is showed. Q Q Q By Lemma 3.2, (K ⊕ K ⊕ K ) > (K ⊕ G [V (H )] ⊕ K ). Since (K ) = 1 |H | n−k−1 1 i k n−k−1 i i i n Q Q (K ⊕ K ⊕ K ), then (K ) is maximal and the theorem holds. 1 |V | n−k−1 i i i n k k k Since K ∈ E ⊂ V , the following result is immediate. n n n Theorem 4.5 Let G be a graph in E . Then 2 2 2(n−k−1) (G) ≤ k (n − k) k(n − 2) and k k(n−1) (n−2)(n−k−1) (G) ≤ k (n − 1) (n − 2) , where the equalities hold if and only if G = K . In the rest of this Section, we consider the minimal mutiplicative Zagreb indices of graphs G k k in V and E . By Proposition 2.2 (ii), G is a tree with n vertices. By Lemma 3.1 and routine n n calculations, we have Theorem 4.6 Let G be a graph in V . Then Y Y 2 n−2 (G) ≥ (n − 1) and (G) ≥ 4 , 1 2 ∼ ∼ where the equalities hold if and only if G S and G P , respectively. = = n n 7 k k Note that P , S ∈ E ⊂ V , then the following theorem is obvious. n n n n Theorem 4.7 Let G be a graph in E . Then Y Y 2 n−2 (G) ≥ (n − 1) and (G) ≥ 4 , 1 2 ∼ ∼ where the equalities hold if and only if G S and G P , respectively. = = n n 5 Extremal graphs with given number of pendant vertices Let G be the set of graphs with p ≥ 2 pendant vertices. In this section, the maximal and minimal multiplicative Zagreb indices of graphs with p pendant vertices in G are determined, and the corresponding extremal graphs shall be characterized in Theorems 5.1 and 5.5. Before exhibiting the main results of the section, we list some notations which will be used in the sequel. Clearly, if G ∈ G , then there be a connected subgraph H with order n−p for which G n 1 can reconstructed by linking p vertices to some vertices H . Especially, since H is connected, it has 1 1 1 2 ∼ ∼ two extremal cases, i.e., H = K and H = T . Let A and A be the two graph sets such that 1 n−p 1 n−p n n its element with the sequence (p, 2, . . . , 2, 1, . . . , 1) and (k + 1, . . . , k + 1, k, . . . , k, 1, . . . , 1), where | {z } | {z } | {z } | {z } | {z } n−p−1 p r n−p−r p 2n − p − 2 = k(n − p) + r with k ≥ 2 and 0 ≤ r ≤ n − p − 1. Let T be a tree, and v ∈ V (T ) with d(v) = k. Note that T − v has k components, for each component associated with v, we call it as a branch of v. We notice that graph G meets |n − n | ≤ 1 for 1 ≤ i, j ≤ n − p, see Fig.2. a i j n−p Since n = p. There are two integers ℓ and t for which p = ℓ(n − p) + t with ℓ ≥ 0 and , ℓ, . . . , ℓ, 1, . . . , 1). 0 ≤ t ≤ n − p − 1. In other words, G has the sequence (ℓ + 1, . . . , ℓ + 1 | {z } | {z } | {z } t n−p−t n p n n n−p 1 2 z}|{ z}|{ z}|{ z }| { 1 v n−p n−p n−p G G a s Figure.2 The graphs G with |n − n | ≤ 1 for 1 ≤ i, j ≤ n − p and G . a i j s Theorem 5.1 Let G be a graph in G . Then Y Y 2t 2(n−p−ℓ) n−1 (n−p−1) (G) ≤ (n + ℓ − p) (n + ℓ − p − 1) and (G) ≤ (n − 1) (n − p − 1) , 1 2 ∼ ∼ where the equalities hold if and only if G = G and G = G , respectively(see, Fig. 2). a s Q Q Proof. Suppose that G ∈ G such that G has the maximum value with respect to and . 1 2 Q Q Q According to properties of for i = 1, 2, if G+e ∈ G , we obtain that (G) < (G+e). Hence i i i 8 the subgraph H of G with order n − p is the complete graph K . Labeling the vertices of H as 1 n−p 1 v , v , . . . , v , let n be the number of pendant vertices who link with v , for i = 1, 2, . . . , n − p. 1 2 n−p i i We firstly show the upper bound of . Assume that there are two vertices v and v of H such that |n − n | ≥ 2. With loss of i j 1 i j generality, set n − n ≥ 2. Let G be the new graph from G by deleting one pendent vertex and i j adding it to v . Note that d ′(v ) = d (v ) − 1 and d ′(v ) = d (v ) + 1. For convenience, we write j G i G i G j G j d instead of d . Observe that ′ 2 2 2 2 (G ) d (v )d (v ) (d(v ) − 1) (d(v ) + 1) ′ ′ i j i j 1 G G Q = = 2 2 2 2 (G) d (v )d (v ) d (v )d (v ) i j i j G G 2 2 2d(v )d(v )(d(v ) − d(v ) − 2) + (d(v ) − 1) + d (v ) + 2d(v ) i j i j i j j = 1 + 2 2 d (v )d (v ) i j d (v ) + 2d(v ) j j ≥ 1 + > 1. 2 2 d (v )d (v ) i j Q Q Consequently, (G ) > (G) which contradicts with the choice of G. Hence, for any pair v 1 1 and v of H , |n − n | ≤ 1. In other words, G G . Clearly, by routine calculation, (G ) = j 1 i j a a 2t 2(n−p−ℓ) (n + ℓ − p) (n + ℓ − p − 1) . Q Q We now verify the upper bound of . In order to obtain the maximum of , it is sufficient 2 2 to show the following claim. Claim 5.2 The number of the vertices in H who possesses pendent vertex is one. Proof. Assume that G has at least two vertices, such as, v and v ( with d(v ) ≥ d(v )), which have i j i j ′′ pendents. Denote by G the graph obtained from G by deleting one pendent of v and adding to v . ′′ ′′ So d (v ) − d (v ) ≥ 2. i j G G According to Proposition 2.3, we observe that ′′ d ′′ (v ) d ′′ (v ) d(v )+1 d(v )−1 i j i j G G ′′ ′′ (G ) d (v ) d (v ) (d(v ) + 1) (d(v ) − 1) G i G j i j = = d (v ) d (v ) d(v ) d(v ) G i G j i j (G) d (v ) d (v ) d(v ) d(v ) G i G j i j > 1(by setting d(v ) , d(v ) − 1). i j Q Q ′′ So, (G ) > (G), a contradiction. Therefore, the claim is holds. 2 2 Clearly, G is the graph such that has maximum if and only if G = G , and G is maximal Q Q graph regarding if and only if G = G . By direct calculation, we have (G ) = (n + ℓ − s a 2 1 2t 2(n−p−ℓ) n−1 (n−p−1) p) (n + ℓ − p − 1) and (G ) = (n − 1) (n − p − 1) . Therefore, we complete the proof. 9 n n n 1 2 1 z }| { z }| { z }| { v v v v 1 2 1 2 |{z} G G Figure.3 The graphs G and G used in Lemma 5.3. 1 2 n n 2 1 n − 1 n + 1 2 1 z }| { z }| { z}|{ z }| { v v v v 2 1 2 1 G G Figure.4 The graphs G and G used in Lemma 5.4 3 4 Lemma 5.3 If G and G are two graphs as shown in Fig. 3, and G is regarded as the graph 1 1 Q Q obtained from G by transferring n branches of v to v . Then (G ) < (G). 2 2 1 1 1 1 Proof. We always suppose d (v ) ≥ d (v ) (If not, exchanging the signs of v and v ). Clearly, G 1 G 2 1 2 d (v ) = n + 2 and d (v ) = n + 2. In view of the property of and Proposition 2.5, we have G 1 1 G 2 2 2 2 (G ) d (v ) d (v ) 1 G 1 G 2 1 1 1 2 2 (G) d (v ) d (v ) G 1 G 2 2 2 (n + n + 2) 2 1 2 2 2 (n + 2) (n + 2) 1 2 < 1. Hence, the proof is complete. Lemma 5.4 Let G and G be two graphs as shown in Fig. 4, and G is considered as the graph 2 2 obtained from G by deleting one branch of v and adding to v . If d (v ) − d (v ) ≥ 2. Then 2 1 G 2 G 1 Q Q (G ) < (G). 1 1 Proof. Note that d (v ) = n + 2 and d (v ) = n + 2. Since d (v ) − d (v ) ≥ 2. According to G 1 1 G 2 2 G 2 G 1 the Proposition 2.4 and the property of , we find that d (v ) d (v ) G 1 G 2 2 2 (G ) d (v ) d (v ) 2 G 1 G 2 2 2 d (v ) d (v ) G 1 G 2 (G) d (v ) d (v ) G 1 G 2 n +3 n +1 1 2 (n + 3) (n + 1) 1 2 n +2 n +2 1 2 (n + 2) (n + 2) 1 2 n +3 n +1+1 1 1 (n + 3) (n + 1 + 1) 1 1 < = 1. n +2 n +1+2 1 1 (n + 2) (n + 1 + 2) 1 1 Therefore, we finish the proof. Theorem 5.5 Let G be a graph in G . Then Y Y 2 2(n−p−1) r(k+1) k(n−p−r) (G) ≥ p 2 and (G) ≥ (k + 1) k , 1 2 1 2 where the equalities hold if and only if G ∈ A and G ∈ A , respectively. n n 10 Q Q ∗ ∗ Proof. Let G be the minimal graph with respect to and , respectively. Obviously, H of G 1 2 Q Q Q is a tree T through (G − e) < (G) for i = 1, 2. We first consider the lower bound of . n−p i i 1 ∗ ∗ If G just has one vertex whose degree is more than three. According to a property of tree, G ∗ ∗ has p pendent vertices. Then G ∈ A . Otherwise, assume that G has at least two vertices with degree more than two (they belong to H .), such as v and v (suppose d ∗(v ) ≥ d ∗(v )). Since 1 i j G i G j two vertices of a tree have a unique path through them. Let P be a maximal path via v and v . t i j We call graph G be the new graph obtained from G by deleting d (v ) − 2 branches of v and 1 G j j Q Q linking to v . In terms of Lemma 5.3, it is easy to deduce that (G ) < (G ). A contradiction i 1 1 1 finish the proof of the part. Next, we discuss the lower bound of . Since G has p pendents. Labeling all vertices of ∗ ∗ ∗ H as v , v , . . . , v , d (v ) ≥ 2 for 1 ≤ i ≤ n − p. We claim that |d (v ) − d (v )| ≤ 1 for 1 1 2 n−p G i G i G j all 1 ≤ i, j ≤ n − p. If not, there exists at least two vertices in H , such as, v and v , such 1 i j 0 0 ∗ ∗ that d (v ) − d (v ) ≥ 2. Considering the new graph G obtained from G by transferring one G i G j 2 0 0 Q Q branch of v to v , by means of Lemma 5.4, we get (G ) < (G ), which is contradicted with i j 2 0 0 2 2 P P the choice of G . Observe that (d (v)) = 2(n−1), and (d (v)) = 2(n−1)−p. G∗ G∗ v∈V (G∗) v∈V (H ) Hence, there exist two integers k, r such that 2(n − 1) − p = k(n − p) + r, where k ≥ 2 and 0 ≤ r ≤ n − p − 1. ∗ 1 ∗ Combining the above discussion, we deduce that G belongs to A for , and G belongs to 2 1 2 A with respect to . From the definition of A and A , through a direct calculation, we have n n n Q Q ∗ 2 2(n−p−1) ∗ r(k+1) k(n−p−r) (G ) = p 2 and (G ) = (k + 1) k . 1 2 Therefore, we finish the proof. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos.11401348 and 11561032), Shandong Provincial Natural Science Foundation of China(No. ZR2019MA012),and supported by Postdoctoral Science Foundation of China. References [1] M. Randi´c, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609- [2] L.B. Kier, L.H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976. [3] L.B. Kier, L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press/Wiley, Letchworth, New York, 1986. [4] L. Pogliani, From molecular connectivity indices to semiempirical connectivity terms: Recent trends in graph theoretical descriptors, Chem. Rev. 100 (2000) 3827-3858. 11 [5] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. [6] B. Bollob´as, P. Erd˝os, Graphs of extremal weights, Ars Combin. 50 (1998) 225-233. [7] B. Bollob´as, P. Erd˝os, A. Sarkar, Extremal graphs for weights, Discrete Math. 200 (1999) 5-19. [8] X.L. Li, Y.T. Shi, A survey on the Randi´c index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127-156. [9] M. Randi´c, The connectivity index 25 years after, J. Mol. Graph. Modell. 20 (2001) 19-35. [10] I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. Total-electron energy of alter- nant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538. [11] S.C. Basak, G.D. Grunwald, G.J. Niemi, Use of graph-theoretic geometric molecular descrip- tors in structure-activity relationships, in: A.T. Balaban (Ed.), From Chemical Topology to Three-Dimensional Geometry, Plenum Press, New York, 1997, pp. 73-116. [12] S.C. Basak, B.D. Gute, G.D. Grunwald, A hierarchical approach to the development of QSAR models using topological, geometrical and quantum chemical parameters, in: J. Devillers, A.T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon & Breach, Amsterdam, 1999, pp. 675-696. [13] R. Todeschini, D. Ballabio, V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors Theory and Applications I, Univ. Kragujevac, Kragujevac, (2010) 72-100. [14] R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010) 359-372. [15] W. Gao, M. Farahani, M. Husin, S. Wang, On the edge-version atom-bond connectivity and geometric arithmetic indices of certain graph operations, Appl. Math. Comp. 308 (2017) 11-17. [16] A.M. Yu, F. Tian, On the spectral radius of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 51 (2004) 97-109. [17] S.C. Li, H.B. Zhou, On the maximum and minimum Zagreb indices of graphs with connectivity at most k, Appl. Math. Lett. 23 (2010) 128-132. [18] X.F. Pan, H.Q. Liu, J.M. Xu, Sharp lower bounds for the general Randi´c index of trees with a given size of matching, MATCH Commun. Math. Comput. Chem. 54 (2005) 465-480. [19] H. Lei, T. Li, Y. Shi, H. Wang, Wiener polarity index and its generalization in trees, MATCH Commun. Math. Comput. Chem. 78(1)(2017), 199-212. [20] J. Ma, Y. Shi, Z. Wang, J. Yue, On Wiener polarity index of bicyclic networks, Sci. Rep. 6(2016), 19066. 12 [21] Y. Shi, Note on two generalizations of the Randi´c index, Appl. Math. Comput. 265(2015), 1019–1025. [22] J. Estes, B. Wei, Sharp bounds of the Zagreb indices of k-trees. J. Comb. Optim. (2014) 27: 271-291. [23] K. Das, I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004) 103-112. [24] I. Gutman, K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83-92. [25] A.L. Hou, S.C. Li, L.Z. Song, B. Wei, Sharp bounds for Zagreb indices of maximal outerplanar graphs, J. Comb. Optim. (2011) 22:252-269. [26] A. Iranmanesh, M.A. Hosseinzadeh, I. Gutman, On multiplicative Zagreb indices of graphs, Iranian J. Math. Chem. 3 (2) (2012) 145-154. [27] B. Borovi´canin, B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016) 208-218. [28] B. Borovi´canin, T.A. Lampert, On the maximum and minimum Zagreb indices of trees with a given number of vertices of maximum degree, MATCH Commun. Math. Comput. Chem. 74 (2015) 81-96. [29] K. Xu, H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 241-256. [30] S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175. [31] J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012) 231-240. [32] Su, G., Xiong, L., Su, X. et al. Some results on the reciprocal sum-degree distance of graphs. J Comb Optim 30, 435¨C446 (2015). https://doi.org/10.1007/s10878-013-9645-5. [33] S. Ji, S. Wang, On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices, J. Math. Anal. Appl., 458 (1) 2018, 21-29. [34] Mart´ınez-P´erez, A., Rodr´ıguez, J.M. New lower bounds for the second variable Zagreb index. J Comb Optim 36, 194¨C210 (2018). https://doi.org/10.1007/s10878-018-0293-7 [35] R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math. 198 (2016) 147- [36] S. Li, Q. Zhao, Sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number, Mathematical and Computer Modelling 54 (2011) 2869-2879. 13 [37] Q. Zhao, S. Li, On the maximum Zagreb indices of graphs with k cut vertices , Acta Appl. Math. (2010) 111 93-106. [38] B. Bollob´as, Modern Graph Theory, Springer-Verlag, 1998. [39] N. Trinajsti´c, Chemical Graph Theory, CRC Press, 1992. [40] Vetr´ık T., Balachandran S., General multiplicative Zagreb indices of trees and unicyclic graphs with given matching number. J Comb Optim (2020). https://doi.org/10.1007/s10878-020- 00643-8. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics arXiv (Cornell University)

The multiplicative Zagreb indices of graphs with given connectivity or number of pendant vertices

Ji, Shengjin; Wang, Shaohui; Muche, Tilahun; Hayat, Sakander
Mathematics , Volume 2021 (1711) – Nov 24, 2017
Download PDF
14 pages

Loading next page...
 
/lp/arxiv-cornell-university/the-multiplicative-zagreb-indices-of-graphs-with-given-connectivity-or-HeeD5cddLC

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

eISSN
ARCH-3343
Publisher site
See Article on Publisher Site

Abstract

For a graph G, the first multiplicative Zagreb index (G) is the product of squares of vertex degrees, and the second multiplicative Zagreb index (G) is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal Π (G) and Π (G) in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most k and p pendant vertices. In addition, Q Q the maximum and minimum values of (G) and (G) are provided. Our results extend and 1 2 enrich some known conclusions. Keywords: Connectivity; Edge connectivity; Extremal bounds; Multiplicative Zagreb indices; Pendant vertices. AMS subject classification: 05C35, 05C38, 05C75, 05C76, 05C09, 05C92 1 Introduction A topological index is a single number which can be used to describe some properties of a molecular graph that is a finite simple graph, representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. In recent decades, these numerical quantities have been found useful for the study of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) and for the structural essence of biological and chemical compounds. The well-known Randi´c index is one of the most important topological indices. In 1975, Randi´c introduced a moleculor quantity of branching index [1], which has been known as the famous Randi´c connectivity index and that is a most useful structural descriptor in QSPR and QSAR, see [2, 3, 4, 5]. Mathematicians have considerable interests in the structural and applied aspects of Randi´c connectivity index, see [6, 7, 8, 9]. Based on the successful considerations, Zagreb indices [10] are introduced as an expected formula for the total π-electron energy of conjugated Corresponding author. Email addresses: [email protected](S. Ji), [email protected](S. Wang), [email protected](T. Muche), [email protected](S. Hayat). arXiv:1711.09014v2 [math.CO] 6 Aug 2021 molecules as follows. X X M (G) = d(u) and M (G) = d(u)d(v), 1 2 u∈V (G) uv∈E(G) where G is a (molecular) graph, uv is a bond between two atoms u and v, and d(u) (resp. d(v)) is the number of atoms that are connected with u (resp. v). Zagreb indices have also been employed as molecular descriptors in QSPR and QSAR, see [11, 12]. Recently, Todeschini et al. (2010) [13, 14] proposed the following multiplicative variants of molecular structure descriptors: Y Y Y Y Y 2 d(u) (G) = d(u) and (G) = d(u)d(v) = d(u) . 1 2 u∈V (G) uv∈E(G) u∈V (G) In the interplay among mathemactics, chemistry and physics, it is not surprising that there are numerous studies of properties of the (multiplicative) Zagreb indices of molecular graphs [15, 16, 17, 18, 19, 20, 21, 22, 32, 34, 40]. In view of these results, researchers are interested in finding upper and lower bounds for multi- plicative Zagreb indices of graphs and characterizing the graphs in which the maximal and minimal index values are attained. In view of the above problems, various mathematical and computational properties of Zagreb indices have been investigated in [23, 24, 25]. Other directions of investigation include studies of relation between multiplicative Zagreb indices and the corresponding invariant of elements of the graph G (vertices, pendant vertices, diameter, maximum degree, girth, cut edge, cut vertex, connectivity, perfect matching). For instance, the first and second multiplicative Zagreb indices for a class of chemical den- drimers are explored by Iranmanesh et al. [26]. Considering trees, unicyclic graphs and bicyclic graphs, Borovi´canin et al. [27] introduced the bounds on Zagreb indices with a fixed domination number. The maximum and minimum Zagreb indices of trees with given number of vertices of maximum degree are proposed by Borovic´anin and Lampert [28]. Xu and Hua [29] introduced a unified approach to characterize maximal and minimal multiplicative Zagreb indices, respectively. Considering the trees of higher dimension, i.e. k-trees, Wang and Wei [30] provided the maximum and minimum values of these indices and the corresponding extremal graphs. Some sharp upper Q Q bounds for -index and -index in terms of graph parameters are investigated by Liu and Zhang 1 2 [31], including the order, size and radius of graphs. Ji and Wang [33] provided the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices. The bounds for the moments and the probability generating function of these indices in a randomly chosen molecular graph with tree structure of given order are studied by Kazemi [35]. Li and Zhao obtained sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number [36, 37]. In light of the information available for multiplicative Zagreb indices, and inspired by above results, in this paper we further investigate these indices of graphs with a given (edge) connec- tivity and number of pendant vertices. We give some basic properties of the first and the second Q Q multiplicative Zagreb indices. The maximum and minimum values of (G) and (G) of graphs 1 2 with given (edge) connectivity at most k and p pendant vertices are provided. In addition, the 2 corresponding extremal graphs are charaterized. In our exposition, we will use the terminology and notations of (chemical) graph theory (see [38, 39]). 2 Preliminaries Let G be a simple connected graph, denoted by G = (V (G), E(G)), in which V = V (G) is vertex set and E = E(G) is edge set. For a vertex v ∈ V (G), the neighborhood of v is the set N(v) = N (v) = {w ∈ V (G), vw ∈ E(G)}, and d (v) (or d(v)) is the degree of v with d (v) = |N(v)|. For G G G i ≥ 0, n denoted the number of vertices of degree i. For S ⊆ V (G) and F ⊆ E(G), we use G[S] for the subgraph of G induced by the vertex set S, G − S for the subgraph induced by V (G) − S and G − F for the subgraph of G obtained by deleting F. If G − S contains at least 2 components, then S is said to be a vertex cut set of G. Similarly, if G − F contains at least 2 components, then F is called an edge cut set. A graph G is said to be k-connected with k ≥ 1, if either G is complete graph K , or it has k+1 at least k + 2 vertices and contains no (k − 1)-vertex cut. The connectivity of G, denoted by κ(G), is defined as the maximal value of k for which a connected graph G is k-connected. Similarly, for k ≥ 1, a graph G is called k-edge-connected if it has at least two vertices and does not contain a (k − 1)-edge cut. The maximal value of k for which a connected graph G is k-edge-connected is said to be the edge connectivity of G, denoted by κ (G). According to the above definitions, the following proposition is obtained. Proposition 2.1 Let G be a graph with n vertices. Then (i) κ(G) ≤ κ (G) ≤ n − 1, (ii) κ(G) = κ (G) = n − 1 if and only if G K . k k Let V be the set of graphs with n vertices and κ(G) ≤ k ≤ n − 1. Denote E by the set of graphs n n with n vertices and κ (G) ≤ k ≤ n − 1. Note that if |V (G)| = n and |E(G)| = n − 1, then G is a tree. Let P and S be special trees: a path and a star of n vertices. The graph K is obtained by n n k k k joining k vertices of K to an isolated vertex, see Fig 1. Then K ∈ E ⊂ V . n−1 n n n n−k−j n−1 Figure.1 The graphs K and G(j, n − k − j) = K ⊕ H ⊕ K . j k n−k−j Q Q Considering the concepts of (G) and (G), the following proposition is routinely obtained. 1 2 3 k k Proposition 2.2 Let e be an edge of a graph G ∈ V (resp. E ). Then n n k k (i) G − e ∈ V (resp. E ), n n Q Q (ii) (G − e) < (G),i=1,2. i i In addition, by elementary calculations, these three statements are deduced. (x+m) Proposition 2.3 For m ≥ 0, F (x) = is monotonically increasing in (0, +∞). 1 x−1 (x−1+m) Proposition 2.4 If m ≥ 0, then F (x) = is a decreasing function in interval (0, +∞). 2 x+m (x+m) 2 2 n Proposition 2.5 F (x) = (x) (n − x) is monotonically increasing for x ∈ [1, ⌊ ⌋]. 3 Lemmas We first provide some lemmas, which are important in proving our main results. Q Q Lemma 3.1 [26] Let T be a tree on n vertices. If T is not P or S , then both (T ) > (S ) n n n 1 1 Q Q and (T ) > (P ) holds. 2 2 Q Q Considering the definitions of (G) and (G), we have the following lemma. 1 2 Lemma 3.2 Let u, v ∈ V (G) such that uv ∈/ E(G). Then Y Y Y Y (G + uv) > (G) and (G + uv) > (G). 1 1 2 2 Given two graphs G and G , if V (G ) ∩ V (G ) = φ, then the join graph G ⊕ G is a graph 1 2 1 2 1 2 with vertex set V (G ) ∪ V (G ) and edge set E(G ) ∪ E(G ) ∪ {uv, u ∈ V (G ), v ∈ V (G )}. 1 2 1 2 1 2 Lemma 3.3 Let G(j, n − k − j) = K ⊕ H ⊕ K be a graph with n vertices, in which K and j k n−k−j j n−k K are cliques, and H is a graph with k vertices, see Fig 2. If k ≥ 1 and 2 ≤ j ≤ , then n−k−j k Y Y (G(j, n − k − j)) < (G(1, n − k − 1)). 1 1 Proof. We consider the graph from G = G(j, n−k−j) to G = G(j−1, n−k−j +1). Note that if 1 2 v ∈ V (H ) in G , then d (v) = d (v); if v ∈ V (K ) in G , then d (v) = d (v) − 1 = j + k − 2; k 2 G G j 2 G G 2 1 2 1 Q Q if v ∈ V (K ) in G , then d (v) = d (v) + 1 = n − j. By the definitions of and , we n−k−j+1 2 G G 2 1 1 2 4 have Q Q Q 2 2 2 d(v) d(v) d(v) (G ) v∈V (K ) v∈V (H ) v∈V (K ) 1 j k n−k−j Q Q Q Q 2 2 2 (G ) d(v) d(v) d(v) 1 v∈V (K ) v∈V (H ) v∈V (K ) j−1 k n−k−j+1 j n−k−j 2 2 (j + k − 1) (n − j − 1) j−1 n−k−j+1 2 2 (j + k − 2) (n − j) j+(k−1) j−1 (j−1)+(k−1) =  . n−k−j+1 (n−j−k+1)+(k−1) n−k−j (n−j−k)+(k−1) n−k Since 2 ≤ j ≤ , we obtain j ≤ n − k − j < n − k − j + 1. By Proposition 2.3 and k ≥ 1, we have Q (G ) < 1, (G ) Q Q that is, (G ) < (G ). 1 2 1 1 We can recursively use this process from G to G , and obtain that 1 2 Y Y Y Y (G(j, n−k−j)) < (G(j−1, n−k−j+1)) < (G(j−2, n−k−j+2)) < · · · < (G(1, n−k−1)). 1 1 1 1 Q Q Therefore, (G(j, n − k − j)) < (G(1, n − k − 1)). Thus, we complete the proof. 1 1 Lemma 3.4 Let G be a connected graph and u, v ∈ V (G). Assume that v , v , . . . , v ∈ N(v)\N(u), 1 2 s 1 ≤ s ≤ d(v). Let G = G − {vv , vv , . . . , vv } + {uv , uv , . . . , uv }. If d(u) ≥ d(v) and u is not 1 2 s 1 2 s adjacent to v, then Y Y (G ) > (G). 2 2 Proof. By the concept of (G), we have d(u) Q d(u) d(u) d(v) d(u)+s (G) d(u) d(v) (d(u)+s) = = . d(v)−s d(u)+s d(v)−s (d(v)−s) (G ) (d(u) + s) (d(v) − s) d(v) d(v) By using d(u) ≥ d(v) > d(v) − s and Proposition 2.4, we obtain (G) < 1, (G ) Q Q which implies that (G ) > (G). This shows the lemma. 2 2 n−k Lemma 3.5 If k ≥ 1 and 2 ≤ j ≤ , we have Y Y (G(j, n − k − j)) < (G(1, n − k − 1)). 2 2 5 Proof. Let V (K ) = {v , v , · · · , v } and V (K ) = {u , u , · · · , u }. Note that vertex set j 1 2 j n−k−j 1 2 n−k−j {v , v , · · · v } ⊂ N(v )∩V (K ). We define a new graph G = G(j, n−k−j)−{v v , v v , . . . , v v }+ 2 3 j 1 j 1 2 1 3 1 j Q Q {u v , u v , . . . , u v }. By d(v ) ≤ d(u ) and Lemma 3.4, we have (G ) ≥ (G(j, n − k − j)). 1 2 1 3 1 j 1 1 2 2 ′ ′′ ′ Note that for G , v has neighbors in V (H ) only. Let G = G + {v u , 2 ≤ i ≤ j, 1 ≤ l ≤ 1 k i l Q Q Q ′ ′′ ′ n − k − j and v u ∈/ E(G )}. By Lemma 3.2, we have (G ) > (G ) ≥ (G). Therefore i l 2 2 2 Q Q ′′ G G(1, n − k − 1) and (G(j, n − k − j)) < (G(1, n − k − 1)). This completes the proof. 2 2 4 Extremal graphs with given connectivity In this section, the maximal and minimal multiplicative Zagreb indices of graphs with connectiv- k k ity at most k in V and E are determined, and the corresponding extremal graphs have been n n characterized in Theorems 4.1 and 4.5. Theorem 4.1 Let G be a graph in V . Then 2 2 2(n−k−1) (G) ≤ k (n − k) k(n − 2) and k k(n−1) (n−2)(n−k−1) (G) ≤ k (n − 1) (n − 2) , where the equalities hold if and only if G = K . Proof. Note that the degree sequence of K is k, n − 2, n − 2, · · · , n − 2, n − 1, n − 1, · · · , n − 1. | {z } | {z } n−k−1 k Q Q By the concepts of (G), (G) and routine calculations, we have 1 2 k 2 2k 2(n−k−1) (K ) = k (n − 1) (n − 2) and k k k(n−1) (n−2)(n−k−1) (K ) = k (n − 1) (n − 2) . Q Q Q Q k k It suffices to prove that (G) ≤ (K ) and (G) ≤ (K ), and the equalities hold if and n n 1 1 2 2 only if G K . n−1 ∼ ∼ If k ≥ n − 1, then G K K , and the theorem is true. If 1 ≤ k ≤ n − 2, then choose = = Q Q a graph G (resp. G ) in V such that (G ) (resp. (G )) is maximal. Since G ≇ K with 1 2 1 2 i n n 1 2 i = 1, 2, then G has a vertex cut set of size k. Let V = {v , v , · · · , v } be the cut vertex set of i i i1 i2 ik G . Denoted ω(G − V ) by the number of components of G − V . In order to prove our theorem, i i i i i we start with several claims. Claim 4.2 ω(G − V ) = 2 with i = 1, 2. i i 6 Proof. We proceed to prove it by a contradiction. Assume that ω(G − V ) ≥ 3 with i = 1, 2. Let i i G , G , · · · , G be the components of G − V . Since ω(G − V ) ≥ 3, then choose vertices 1 2 i i i i ω(G −V ) i i u ∈ V (G ) and v ∈ V (G ). Then V is still a k-vertex cut set of G + uv. By Lemma 3.2, we have 1 2 i i Q Q (G + uv) > (G ), a contradiction to the choice of G . Thus, this claim is proved. i i i i i Without loss of generality, suppose that G − V contains only two connected components, i i denoted by G and G . i1 i2 Claim 4.3 The induced graphs on V (G ) ∪ V and V (G ) ∪ V in G are complete subgraphs. i1 i i2 i i Proof. We use a contradiction to show it. Suppose that G [V (G ) ∪ V ] is not a complete subgraph i i1 i of G . Then there exists an edge uv ∈/ G [V (G ) ∪ V ]. Since G [V (G ) ∪ V ] + uv ∈ V , by i i i1 i i i1 i n,k Q Q Lemma 3.2, we have (G [V (G )∪V ]+uv) > (G [V (G )∪V ]), which is a contadiction. This i i1 i i i1 i i i shows the claim. By the above claims, we see that G and G are complete subgraph of G . Let G = K and i1 i2 i i1 n ′′ ′ ′′ G = K . Then we have G = K ⊕ G [V ] ⊕ K . i2 n i n i i n ′ ′′ Claim 4.4 Either n = 1 or n = 1. ′ ′′ ′ ′′ Proof. On the contrary, assume that n , n ≥ 2. Without loss of generility, n ≤ n . For (G), by Q Q ′ ′ Lemmas 3.3 and 3.5, we have a new graph G = K ⊕G [V ]⊕K such that (G ) > (G ) i 1 i i n−k−1 i i i i k ′ ′′ and G ∈ V . This is a contradition to the choice of G . Thus, either n = 1 or n = 1, and this i i claim is showed. Q Q Q By Lemma 3.2, (K ⊕ K ⊕ K ) > (K ⊕ G [V (H )] ⊕ K ). Since (K ) = 1 |H | n−k−1 1 i k n−k−1 i i i n Q Q (K ⊕ K ⊕ K ), then (K ) is maximal and the theorem holds. 1 |V | n−k−1 i i i n k k k Since K ∈ E ⊂ V , the following result is immediate. n n n Theorem 4.5 Let G be a graph in E . Then 2 2 2(n−k−1) (G) ≤ k (n − k) k(n − 2) and k k(n−1) (n−2)(n−k−1) (G) ≤ k (n − 1) (n − 2) , where the equalities hold if and only if G = K . In the rest of this Section, we consider the minimal mutiplicative Zagreb indices of graphs G k k in V and E . By Proposition 2.2 (ii), G is a tree with n vertices. By Lemma 3.1 and routine n n calculations, we have Theorem 4.6 Let G be a graph in V . Then Y Y 2 n−2 (G) ≥ (n − 1) and (G) ≥ 4 , 1 2 ∼ ∼ where the equalities hold if and only if G S and G P , respectively. = = n n 7 k k Note that P , S ∈ E ⊂ V , then the following theorem is obvious. n n n n Theorem 4.7 Let G be a graph in E . Then Y Y 2 n−2 (G) ≥ (n − 1) and (G) ≥ 4 , 1 2 ∼ ∼ where the equalities hold if and only if G S and G P , respectively. = = n n 5 Extremal graphs with given number of pendant vertices Let G be the set of graphs with p ≥ 2 pendant vertices. In this section, the maximal and minimal multiplicative Zagreb indices of graphs with p pendant vertices in G are determined, and the corresponding extremal graphs shall be characterized in Theorems 5.1 and 5.5. Before exhibiting the main results of the section, we list some notations which will be used in the sequel. Clearly, if G ∈ G , then there be a connected subgraph H with order n−p for which G n 1 can reconstructed by linking p vertices to some vertices H . Especially, since H is connected, it has 1 1 1 2 ∼ ∼ two extremal cases, i.e., H = K and H = T . Let A and A be the two graph sets such that 1 n−p 1 n−p n n its element with the sequence (p, 2, . . . , 2, 1, . . . , 1) and (k + 1, . . . , k + 1, k, . . . , k, 1, . . . , 1), where | {z } | {z } | {z } | {z } | {z } n−p−1 p r n−p−r p 2n − p − 2 = k(n − p) + r with k ≥ 2 and 0 ≤ r ≤ n − p − 1. Let T be a tree, and v ∈ V (T ) with d(v) = k. Note that T − v has k components, for each component associated with v, we call it as a branch of v. We notice that graph G meets |n − n | ≤ 1 for 1 ≤ i, j ≤ n − p, see Fig.2. a i j n−p Since n = p. There are two integers ℓ and t for which p = ℓ(n − p) + t with ℓ ≥ 0 and , ℓ, . . . , ℓ, 1, . . . , 1). 0 ≤ t ≤ n − p − 1. In other words, G has the sequence (ℓ + 1, . . . , ℓ + 1 | {z } | {z } | {z } t n−p−t n p n n n−p 1 2 z}|{ z}|{ z}|{ z }| { 1 v n−p n−p n−p G G a s Figure.2 The graphs G with |n − n | ≤ 1 for 1 ≤ i, j ≤ n − p and G . a i j s Theorem 5.1 Let G be a graph in G . Then Y Y 2t 2(n−p−ℓ) n−1 (n−p−1) (G) ≤ (n + ℓ − p) (n + ℓ − p − 1) and (G) ≤ (n − 1) (n − p − 1) , 1 2 ∼ ∼ where the equalities hold if and only if G = G and G = G , respectively(see, Fig. 2). a s Q Q Proof. Suppose that G ∈ G such that G has the maximum value with respect to and . 1 2 Q Q Q According to properties of for i = 1, 2, if G+e ∈ G , we obtain that (G) < (G+e). Hence i i i 8 the subgraph H of G with order n − p is the complete graph K . Labeling the vertices of H as 1 n−p 1 v , v , . . . , v , let n be the number of pendant vertices who link with v , for i = 1, 2, . . . , n − p. 1 2 n−p i i We firstly show the upper bound of . Assume that there are two vertices v and v of H such that |n − n | ≥ 2. With loss of i j 1 i j generality, set n − n ≥ 2. Let G be the new graph from G by deleting one pendent vertex and i j adding it to v . Note that d ′(v ) = d (v ) − 1 and d ′(v ) = d (v ) + 1. For convenience, we write j G i G i G j G j d instead of d . Observe that ′ 2 2 2 2 (G ) d (v )d (v ) (d(v ) − 1) (d(v ) + 1) ′ ′ i j i j 1 G G Q = = 2 2 2 2 (G) d (v )d (v ) d (v )d (v ) i j i j G G 2 2 2d(v )d(v )(d(v ) − d(v ) − 2) + (d(v ) − 1) + d (v ) + 2d(v ) i j i j i j j = 1 + 2 2 d (v )d (v ) i j d (v ) + 2d(v ) j j ≥ 1 + > 1. 2 2 d (v )d (v ) i j Q Q Consequently, (G ) > (G) which contradicts with the choice of G. Hence, for any pair v 1 1 and v of H , |n − n | ≤ 1. In other words, G G . Clearly, by routine calculation, (G ) = j 1 i j a a 2t 2(n−p−ℓ) (n + ℓ − p) (n + ℓ − p − 1) . Q Q We now verify the upper bound of . In order to obtain the maximum of , it is sufficient 2 2 to show the following claim. Claim 5.2 The number of the vertices in H who possesses pendent vertex is one. Proof. Assume that G has at least two vertices, such as, v and v ( with d(v ) ≥ d(v )), which have i j i j ′′ pendents. Denote by G the graph obtained from G by deleting one pendent of v and adding to v . ′′ ′′ So d (v ) − d (v ) ≥ 2. i j G G According to Proposition 2.3, we observe that ′′ d ′′ (v ) d ′′ (v ) d(v )+1 d(v )−1 i j i j G G ′′ ′′ (G ) d (v ) d (v ) (d(v ) + 1) (d(v ) − 1) G i G j i j = = d (v ) d (v ) d(v ) d(v ) G i G j i j (G) d (v ) d (v ) d(v ) d(v ) G i G j i j > 1(by setting d(v ) , d(v ) − 1). i j Q Q ′′ So, (G ) > (G), a contradiction. Therefore, the claim is holds. 2 2 Clearly, G is the graph such that has maximum if and only if G = G , and G is maximal Q Q graph regarding if and only if G = G . By direct calculation, we have (G ) = (n + ℓ − s a 2 1 2t 2(n−p−ℓ) n−1 (n−p−1) p) (n + ℓ − p − 1) and (G ) = (n − 1) (n − p − 1) . Therefore, we complete the proof. 9 n n n 1 2 1 z }| { z }| { z }| { v v v v 1 2 1 2 |{z} G G Figure.3 The graphs G and G used in Lemma 5.3. 1 2 n n 2 1 n − 1 n + 1 2 1 z }| { z }| { z}|{ z }| { v v v v 2 1 2 1 G G Figure.4 The graphs G and G used in Lemma 5.4 3 4 Lemma 5.3 If G and G are two graphs as shown in Fig. 3, and G is regarded as the graph 1 1 Q Q obtained from G by transferring n branches of v to v . Then (G ) < (G). 2 2 1 1 1 1 Proof. We always suppose d (v ) ≥ d (v ) (If not, exchanging the signs of v and v ). Clearly, G 1 G 2 1 2 d (v ) = n + 2 and d (v ) = n + 2. In view of the property of and Proposition 2.5, we have G 1 1 G 2 2 2 2 (G ) d (v ) d (v ) 1 G 1 G 2 1 1 1 2 2 (G) d (v ) d (v ) G 1 G 2 2 2 (n + n + 2) 2 1 2 2 2 (n + 2) (n + 2) 1 2 < 1. Hence, the proof is complete. Lemma 5.4 Let G and G be two graphs as shown in Fig. 4, and G is considered as the graph 2 2 obtained from G by deleting one branch of v and adding to v . If d (v ) − d (v ) ≥ 2. Then 2 1 G 2 G 1 Q Q (G ) < (G). 1 1 Proof. Note that d (v ) = n + 2 and d (v ) = n + 2. Since d (v ) − d (v ) ≥ 2. According to G 1 1 G 2 2 G 2 G 1 the Proposition 2.4 and the property of , we find that d (v ) d (v ) G 1 G 2 2 2 (G ) d (v ) d (v ) 2 G 1 G 2 2 2 d (v ) d (v ) G 1 G 2 (G) d (v ) d (v ) G 1 G 2 n +3 n +1 1 2 (n + 3) (n + 1) 1 2 n +2 n +2 1 2 (n + 2) (n + 2) 1 2 n +3 n +1+1 1 1 (n + 3) (n + 1 + 1) 1 1 < = 1. n +2 n +1+2 1 1 (n + 2) (n + 1 + 2) 1 1 Therefore, we finish the proof. Theorem 5.5 Let G be a graph in G . Then Y Y 2 2(n−p−1) r(k+1) k(n−p−r) (G) ≥ p 2 and (G) ≥ (k + 1) k , 1 2 1 2 where the equalities hold if and only if G ∈ A and G ∈ A , respectively. n n 10 Q Q ∗ ∗ Proof. Let G be the minimal graph with respect to and , respectively. Obviously, H of G 1 2 Q Q Q is a tree T through (G − e) < (G) for i = 1, 2. We first consider the lower bound of . n−p i i 1 ∗ ∗ If G just has one vertex whose degree is more than three. According to a property of tree, G ∗ ∗ has p pendent vertices. Then G ∈ A . Otherwise, assume that G has at least two vertices with degree more than two (they belong to H .), such as v and v (suppose d ∗(v ) ≥ d ∗(v )). Since 1 i j G i G j two vertices of a tree have a unique path through them. Let P be a maximal path via v and v . t i j We call graph G be the new graph obtained from G by deleting d (v ) − 2 branches of v and 1 G j j Q Q linking to v . In terms of Lemma 5.3, it is easy to deduce that (G ) < (G ). A contradiction i 1 1 1 finish the proof of the part. Next, we discuss the lower bound of . Since G has p pendents. Labeling all vertices of ∗ ∗ ∗ H as v , v , . . . , v , d (v ) ≥ 2 for 1 ≤ i ≤ n − p. We claim that |d (v ) − d (v )| ≤ 1 for 1 1 2 n−p G i G i G j all 1 ≤ i, j ≤ n − p. If not, there exists at least two vertices in H , such as, v and v , such 1 i j 0 0 ∗ ∗ that d (v ) − d (v ) ≥ 2. Considering the new graph G obtained from G by transferring one G i G j 2 0 0 Q Q branch of v to v , by means of Lemma 5.4, we get (G ) < (G ), which is contradicted with i j 2 0 0 2 2 P P the choice of G . Observe that (d (v)) = 2(n−1), and (d (v)) = 2(n−1)−p. G∗ G∗ v∈V (G∗) v∈V (H ) Hence, there exist two integers k, r such that 2(n − 1) − p = k(n − p) + r, where k ≥ 2 and 0 ≤ r ≤ n − p − 1. ∗ 1 ∗ Combining the above discussion, we deduce that G belongs to A for , and G belongs to 2 1 2 A with respect to . From the definition of A and A , through a direct calculation, we have n n n Q Q ∗ 2 2(n−p−1) ∗ r(k+1) k(n−p−r) (G ) = p 2 and (G ) = (k + 1) k . 1 2 Therefore, we finish the proof. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos.11401348 and 11561032), Shandong Provincial Natural Science Foundation of China(No. ZR2019MA012),and supported by Postdoctoral Science Foundation of China. References [1] M. Randi´c, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609- [2] L.B. Kier, L.H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976. [3] L.B. Kier, L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press/Wiley, Letchworth, New York, 1986. [4] L. Pogliani, From molecular connectivity indices to semiempirical connectivity terms: Recent trends in graph theoretical descriptors, Chem. Rev. 100 (2000) 3827-3858. 11 [5] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. [6] B. Bollob´as, P. Erd˝os, Graphs of extremal weights, Ars Combin. 50 (1998) 225-233. [7] B. Bollob´as, P. Erd˝os, A. Sarkar, Extremal graphs for weights, Discrete Math. 200 (1999) 5-19. [8] X.L. Li, Y.T. Shi, A survey on the Randi´c index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127-156. [9] M. Randi´c, The connectivity index 25 years after, J. Mol. Graph. Modell. 20 (2001) 19-35. [10] I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. Total-electron energy of alter- nant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538. [11] S.C. Basak, G.D. Grunwald, G.J. Niemi, Use of graph-theoretic geometric molecular descrip- tors in structure-activity relationships, in: A.T. Balaban (Ed.), From Chemical Topology to Three-Dimensional Geometry, Plenum Press, New York, 1997, pp. 73-116. [12] S.C. Basak, B.D. Gute, G.D. Grunwald, A hierarchical approach to the development of QSAR models using topological, geometrical and quantum chemical parameters, in: J. Devillers, A.T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon & Breach, Amsterdam, 1999, pp. 675-696. [13] R. Todeschini, D. Ballabio, V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors Theory and Applications I, Univ. Kragujevac, Kragujevac, (2010) 72-100. [14] R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010) 359-372. [15] W. Gao, M. Farahani, M. Husin, S. Wang, On the edge-version atom-bond connectivity and geometric arithmetic indices of certain graph operations, Appl. Math. Comp. 308 (2017) 11-17. [16] A.M. Yu, F. Tian, On the spectral radius of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 51 (2004) 97-109. [17] S.C. Li, H.B. Zhou, On the maximum and minimum Zagreb indices of graphs with connectivity at most k, Appl. Math. Lett. 23 (2010) 128-132. [18] X.F. Pan, H.Q. Liu, J.M. Xu, Sharp lower bounds for the general Randi´c index of trees with a given size of matching, MATCH Commun. Math. Comput. Chem. 54 (2005) 465-480. [19] H. Lei, T. Li, Y. Shi, H. Wang, Wiener polarity index and its generalization in trees, MATCH Commun. Math. Comput. Chem. 78(1)(2017), 199-212. [20] J. Ma, Y. Shi, Z. Wang, J. Yue, On Wiener polarity index of bicyclic networks, Sci. Rep. 6(2016), 19066. 12 [21] Y. Shi, Note on two generalizations of the Randi´c index, Appl. Math. Comput. 265(2015), 1019–1025. [22] J. Estes, B. Wei, Sharp bounds of the Zagreb indices of k-trees. J. Comb. Optim. (2014) 27: 271-291. [23] K. Das, I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004) 103-112. [24] I. Gutman, K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83-92. [25] A.L. Hou, S.C. Li, L.Z. Song, B. Wei, Sharp bounds for Zagreb indices of maximal outerplanar graphs, J. Comb. Optim. (2011) 22:252-269. [26] A. Iranmanesh, M.A. Hosseinzadeh, I. Gutman, On multiplicative Zagreb indices of graphs, Iranian J. Math. Chem. 3 (2) (2012) 145-154. [27] B. Borovi´canin, B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016) 208-218. [28] B. Borovi´canin, T.A. Lampert, On the maximum and minimum Zagreb indices of trees with a given number of vertices of maximum degree, MATCH Commun. Math. Comput. Chem. 74 (2015) 81-96. [29] K. Xu, H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 241-256. [30] S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175. [31] J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012) 231-240. [32] Su, G., Xiong, L., Su, X. et al. Some results on the reciprocal sum-degree distance of graphs. J Comb Optim 30, 435¨C446 (2015). https://doi.org/10.1007/s10878-013-9645-5. [33] S. Ji, S. Wang, On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices, J. Math. Anal. Appl., 458 (1) 2018, 21-29. [34] Mart´ınez-P´erez, A., Rodr´ıguez, J.M. New lower bounds for the second variable Zagreb index. J Comb Optim 36, 194¨C210 (2018). https://doi.org/10.1007/s10878-018-0293-7 [35] R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math. 198 (2016) 147- [36] S. Li, Q. Zhao, Sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number, Mathematical and Computer Modelling 54 (2011) 2869-2879. 13 [37] Q. Zhao, S. Li, On the maximum Zagreb indices of graphs with k cut vertices , Acta Appl. Math. (2010) 111 93-106. [38] B. Bollob´as, Modern Graph Theory, Springer-Verlag, 1998. [39] N. Trinajsti´c, Chemical Graph Theory, CRC Press, 1992. [40] Vetr´ık T., Balachandran S., General multiplicative Zagreb indices of trees and unicyclic graphs with given matching number. J Comb Optim (2020). https://doi.org/10.1007/s10878-020- 00643-8.

Journal

Mathematics – arXiv (Cornell University)

Published: Nov 24, 2017

There are no references for this article.