An extension of the essential graph of a ring (Q6561502)

Graphs from rings have attracted many authors for the past four decades. Right from the notion of zerodivisor graph of commutative rings, several graph constructions are made. Through the tools of graph theory several interlinking connections between rings and graphs are explored. In this paper, authors introduce a graph called extended essential graph of a commutative ring and study its propertis. Let \(R\) be a commutative ring with non-zero identity, and \(E(R) = \{p\in R|\mathrm{ann}_R(pq)\leq_e R, \text{~for some~} q\in R^*\}\). The extended essential graph, denoted by \(E_gG(R)\) is an undirected simple graph with the vertex set \(E(R)^* = E(A)\setminus \{0\}\). Two distinct vertices \(r,s\in E(R)^*\) are adjacent if and only if \(\mathrm{ann}_R(rs) \leq_e R\). In this paper, authors prove that \(E_gG(R)\) is connected with \(\mathrm{diam}(E_gG(R)) \leq 3\) and if \(E_gG(R)\) contains a cycle, then the grith \(\mathrm{gr}(E_gG(R)) \leq 4\). Furthermore, it is established that if \(R\) is an Artinian commutative ring, then the chromatic number \(\omega(E_gG(R)) = \chi(E_gG(R)) = |N(R)^*|+|Max(R)|\).