Extended total graph associated with finite commutative rings (Q6644848)

For a commutative ring \(R\) with identity \(1\not=0\), denote by \(Z(R)\) the set of zero divisors. The total graph of \(R\), denoted by \(T_\Gamma(R)\), is a simple graph in which all elements of \(R\) are vertices and, any two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x+y\in Z(R)\). In this paper, the authors define an extension of the total graph, denoted by \(T(\Gamma^e(R))\) with vertex set \(Z(R)\), in which two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x + y\in Z^*(R)=:Z(R)\setminus \{0\}\). The main aim of the present paper is to characterize all the finite commutative rings whose \(T(\Gamma^e(R))\) has clique numbers \(1, 2,\) and \(3\) respectively. They also characterize finite commutative nonlocal rings \(R\) for which the corresponding graph \(T(\Gamma^e(R))\) has the clique number \(4\). The more difficult local case is yet to be determined.\N\NReviewer's remark: There exist several misprints, e.g., in references 9 and 10, Ji. Guo should be replaced by the correct name Jin Guo for two times.