ON TOTAL GRAPHS RELATED TO COSINGULAR SUBMODULE

Amirmohammad Momeni Kohestani; Yahya Talebi

doi:10.5269/bspm.78488

Amirmohammad Momeni Kohestani Department of Mathematics, University of Mazandaran, Babolsar, Iran
Y. Talebi

Abstract

Let $W$ be a unitary right $A$-module, where $A$ is a ring with identity. In this paper, we will introduce a new total graph that is related to cosingular submodule. The cosingular submodule, denoted by $Z^{*}(W)$, is defined as the set of all elements $w \in W$ such that $wA \ll W$. The total graph of a module W with respect to $Z^{*}(W)$, denoted by $T(\Gamma(W))$, is defined as an undirect simple graph with all elements of $W$ as vertices and two vertices $w$ and $k$ are adjacent, written as $w$ adj $k$, if and only if $w+k \in Z^{*}(W)$. We investigate some results for connectivity, completeness and planarity of these graphs and their induced subgraphs.

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