Co-maximal signed graphs of commutative rings

Authors: DEEPA SINHA, ANITA KUMARI RAO

Abstract: Let $ \Gamma(R)$ be a graph with element of $R$ (finite commutative ring with unity) as vertices, where two vertices $a$ and $b$ are adjacent if and only if $Ra+Rb = R$. In this paper, we characterize the rings for which a co-maximal meet signed graph $ \Gamma_{\Sigma}(R)$, a co-maximal join signed graph $ \Gamma_{\Sigma}^{\vee}(R)$, a co-maximal ring sum signed graph $ \Gamma_{\Sigma}^{\oplus}(R)$, their negation signed graphs $ \eta(\Gamma_{\Sigma}(R))$, $ \eta(\Gamma_{\Sigma}^{\vee}(R))$, $ \eta(\Gamma_{\Sigma}^{\oplus}(R))$ respectively and their line signed graphs are balanced, clusterable, and sign-compatible.

Keywords: Finite commutative ring, maximal ideal, co-maximal graph, balanced signed graph, co-maximal meet signed graph, co-maximal join signed graph, co-maximal ring sum signed graph

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