Discrete Applied Mathematics 159 (2011) 69–78
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
On the revised Szeged index
Rundan Xing, Bo Zhou
∗
Department of Mathematics, South China Normal University, Guangzhou 510631, PR China
a r t i c l e i n f o
Article history:
Received 23 July 2009
Received in revised form 12 July 2010
Accepted 29 September 2010
Available online 27 October 2010
Keywords:
Distance (in graph)
Wiener index
Szeged index
Revised Szeged index
Unicyclic graph
a b s t r a c t
We give bounds for the revised Szeged index, and determine the n-vertex unicyclic graphs
with the smallest, the second-smallest and the third-smallest revised Szeged indices for
n ≥ 5, and the n-vertex unicyclic graphs with the kth-largest revised Szeged indices for
all k up to 3 for n = 5, to 5 for n = 6, to 6 for n = 7, to 7 for n = 8, and to
n
2
+ 4 for
n ≥ 9. We also determine the n-vertex unicyclic graphs of cycle length r, 3 ≤ r ≤ n, with
the smallest and the largest revised Szeged indices.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Topological indices are graph invariants used in theoretical chemistry to encode molecules for the design of chemical
compounds with given physicochemical properties or given pharmacological and biological activities [25]. In this paper, we
consider a topological index named the revised Szeged index, which is closely related to two other topological indices, the
Wiener index and the Szeged index.
Let G be a simple connected graph with vertex set V (G) and edge set E(G). Let d
G
(u, v) be the distance between vertices
u and v in G. The Wiener index of G is defined as [7]
W (G) =
−
{u,v}⊆V (G)
d
G
(u, v).
It has been thoroughly studied; see, e.g., [5,6,1]. Besides of interest for the use in chemistry, it was also independently studied
as of relevance for social science, architecture, and graph theory [22]. If e is an edge of G connecting vertices u and v, then
we write e = uv or e = vu. For e = uv ∈ E(G), let n
1
(e|G) and n
2
(e|G) be respectively the number of vertices of G lying
closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. If G is a tree,
then it is known that W (G) =
∑
e∈E(G)
n
1
(e|G)n
2
(e|G); see [26]. Gutman [3] introduced, as a generalization of this relation,
a graph invariant named the Szeged index. The Szeged index of a connected graph G is defined as [3]
Sz(G) =
−
e∈E(G)
n
1
(e|G)n
2
(e|G).
The Szeged index has received much attention for both its mathematical and computational properties; see, e.g., [11,4,17,
2,27,24], and its various applications in modeling physicochemical properties as well as physiological activities of organic
compounds acting as drugs or possessing pharmacological activity; see [14]. Recent results on the Szeged index may be
∗
Corresponding author.
E-mail address: zhoubo@scnu.edu.cn (B. Zhou).
0166-218X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2010.09.010