A class of graphs with a few well-covered members (Q2041119)

Several graphs are associated with commutative rings and they are used as combinatorial tools to study algebraic properties of commutative rings. Let \(R\) be a commutative ring with identity \(1\neq 0.\) The total graph \(T(R)\) of \(R\) is the simple undirected graph with vertex set \(R\) and two distinct vertices \(x\) and \(y\) are adjacent in \(T(R)\) whenever \(x+y\) is a zerodivisor of \(R.\) A graph is said to be well covered if its all maximal independent sets have the same number of vertices. In this paper, authors characterized all finite commutative rings \(R\) for which the total graph \(T(R)\) is well covered. In fact it is proved that \(T (R)\) is well-covered if and only if either \(R\) is local or \(2(=1+1)\) is a zero-divisor of \(R.\)