Results Math 72 (2017), 193–204
2016 Springer International Publishing
published online August 27, 2016
Results in Mathematics
The Total Graphs of Finite Commutative
David Dolˇzan and Polona Oblak
Abstract. Anderson and Badawi (J Algebra 320(7):2706–2719, 2008)char-
acterized all commutative rings having total graphs without any 3-cycles.
In this paper we expand those results to the semiring setting and obtain
the characterization of ﬁnite commutative semirings having total graphs
without any 3-cycles.
Mathematics Subject Classiﬁcation. 16Y60, 05C25.
Keywords. Finite commutative semiring, zero-divisor, total graph.
Recently, a lot of study of algebraic structures has been explored via the graph
theoretic approach. One of the ﬁrst tools used in this research was the zero-
divisor graph (see e.g. [3,4,6]). In , Anderson and Badawi introduced the
notion of the total graph of a commutative ring R as the graph with all elements
of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent
if and only if x + y is a zero-divisor in R. In many ways, the total graph
reveals more of the essential properties of the ring than the zero-divisor graph,
since it engages both ring operations instead of studying only multiplication.
Anderson and Badawi studied certain graph theoretical parameters of the total
graph of a commutative ring such as diameter and girth. They proved that the
total graph of a ﬁnite commutative ring is connected if and only if the set of
zero-divisors does not form an ideal. Also, they characterised all commutative
rings having total graphs without any 3-cycles.
Since then, many other authors have been studying total graphs. In 
Akbari et al. proved that if the total graph of a ﬁnite commutative ring is
connected then it is also Hamiltonian. In  Maimani et al. gave the necessary