ON THE SUBTRACTIVE SUBSEMIMODULE-BASED GRAPH OF SEMIMODULES

Abstract

Let $M$ be a semimodule over a commutative semiring $R$ and $K$ be a subtractive subsemimodule of $M$ with $K^{\ast }=K\setminus \{0\}$. The subtractive subsemimodule-based graph of $M$ is defined as the simple undirected graph $\mathrm{\Omega }={\mathrm{\Gamma }}_{K^{\ast }}(M)$ with vertex set $V(\mathrm{\Omega })=\{v\in M\setminus K:v+v^{\prime }\in K^{\ast }\text{ for some }v\ne v^{\prime }\in M\setminus K\}$, and two distinct vertices $m$ and $n$ are adjacent if and only if $m+n\in K^{\ast }$. In this paper, we study the interplay between semimodule properties and the properties of the graph. Among other results, we compute the diameter and the girth of ${\mathrm{\Gamma }}_{K^{\ast }}(M)$.