\(\mathcal C\)-consistency in signed total graphs of commutative rings (Q2821105)

The authors study signed total graph of a commutative ring. Let \(R\) be a commutative ring, \(Z(R)\) its set of all zerodivisors, \(U(R)\) be its group of units and \(\mathrm{Reg}(R) = R - Z(R)\) be its set of regular elements. The total graph (introduced by \textit{D. F. Anderson} and \textit{A. Badawi} [J. Algebra 320, No. 7, 2706--2719 (2008; Zbl 1158.13001)]), denoted by \(T(\Gamma(R))\), is a graph whose vertices are the elements of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent iff \(x + y \in Z(R)\).NEWLINENEWLINE\textit{P. Sharma} and \textit{M. Acharya} [Graphs Comb. 32, No. 4, 1585--1597 (2016; Zbl 1342.05060)] have introduced the concept of signed total graph.NEWLINENEWLINEA signed total graph is an ordered pair \(T_{\Sigma}(\Gamma (R)) = (T(\Gamma(R)), \sigma)\), where \(T(\Gamma(R))\) is the total graph of a commutative ring \(R\) and an edge \((a, b)\) of \(T(\Gamma(R)), \sigma\) is defined as NEWLINE\[NEWLINE\sigma(a, b) = \begin{cases} +, & \text{if}\,\, a \in Z(R) \text{ or } b \in Z(R); \\ -, & \text{ otherwise.} \end{cases}NEWLINE\]NEWLINE The following characterizations are proved.NEWLINENEWLINETheorem. Let \(R\) be a commutative ring with unity such that \(Z(R)\) is an ideal of \(R\). Then \(T_{\Sigma}(\Gamma (R))\) is \(\mathcal{C}\) consistent if only if \(2 \notin Z(R)\). Let \(R\) be a commutative ring with unity such that \(Z(R)\) is not an ideal of \(R\). Then \(T_{\Sigma}(\Gamma (R))\) is \(\mathcal{C}\)-consistent if and only if \(R \cong \prod_{i = 1}^t \mathbb{F}_i \), where each \(\mathbb{F}_i\) is a field of characteristic 2.NEWLINENEWLINESome examples of rings \(R\) are given to show that the associated \(T_{\Sigma}(\Gamma (R))\) is \(\mathcal{C}\)-consistent and not \(\mathcal{C}\)-consistent. Also it is shown that for a finite commutative ring, the total signed graph is always sign-compatible.