期刊名称：AKCE International Journal of Graphs and Combinatorics
印刷版ISSN：0972-8600
出版年度：2016
卷号：13
期号：3
页码：290-298
DOI：10.1016/j.akcej.2016.11.009
语种：English
出版社：Elsevier
摘要：Abstract There are so many ways to construct graphs from algebraic structures. Most popular constructions are Cayley graphs, commuting graphs and non-commuting graphs from finite groups and zero-divisor graphs and total graphs from commutative rings. For a commutative ring R with non-zero identity, we denote the set of zero-divisors and unit elements of R by Z ( R ) and U ( R ) , respectively. One of the associated graphs to a ring R is the zero-divisor graph; it is a simple graph with vertex set Z ( R ) ∖ { 0 } , and two vertices x and y are adjacent if and only if x y = 0 . This graph was first introduced by Beck, where all the elements of R are considered as the vertices. Anderson and Badawi, introduced the total graph of R , as the simple graph with all elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z ( R ) . For a given graph G , the concept of connectedness, diameter and girth are always of great interest. Several authors extensively studied about the zero-divisor and total graphs from commutative rings. In this paper, we present a survey of results obtained with regard to distances in zero-divisor and total graphs.
