Reduction graph and its application on algebraic graphs

Abstract

Evans, et al. (J. Graph Theory 18 (1994), 801--815), defined an equivalence relation $\sim $ on the set of vertices of a simple graph $G$ by taking $a\sim b$ if and only if their open neighborhoods are the same. They introduced a new graph $G_{red }={G}/{\sim } $, reduction graph of $G$, as follows. The vertices are $V (G_{red })=\{[a]: a \in V (G) \}$, and two equivalence classes $[a]$ and $[b]$ are adjacent if and only if $a$ and $b$ are adjacent in $G$. Recently, Anderson and LaGrange defined some equivalence relations on the set of vertices of the zero-divisor graph of a commutative ring, one of which yields the reduction graph of the zero-divisor graph. In this paper, we state some basic graph theoretic properties of $G_{red }$ and study the relations between some properties of graph $G$ and its subgraph, $G_{red }$, such as the chromatic number, clique number, girth and diameter. Moreover, we study the reduction graph of some algebraic graphs, such as the comaximal graph, zero-divisor graph and Cayley graph of a commutative ring. Among other results, we show that, for every commutative ring $R$, $\Gamma _2(R)_{red } \simeq \Gamma _1(\mathbb {Z}_2^n)$, where $\Gamma _1(\mathbb {Z}_2^n)$ is the zero-divisor graph of the Boolean ring $\mathbb {Z}_2^n$, $\Gamma _2(R)$ is the comaximal graph of $R$ and $n=|Max (R)|$.