Complement of the reduced non-zero component graph of free semimodules

Abstract

Let \(\mathbb{M}\) be a finitely generated free semimodule over a semiring \(\mathbb{S}\) with identity having invariant basis number property with a basis α = {α₁,…, α_k}. The complement \(\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)\) of the reduced non-zero component graph \({\Gamma ^ * }\left(\mathbb{M}\right)\) of \(\mathbb{M}\), is the simple undirected graph with \(V = {\mathbb{M}^ * }\backslash \left\{ {\sum\limits_{i = 1}^k {{c_i}} {\alpha _i}:{c_i} \ne 0\,\,\forall \,\,i} \right\}\) as the vertex set and such that there is an edge between two distinct vertices \(a = \sum\limits_{i = 1}^k {{a_i}{\alpha _i}} \) and \(b = \sum\limits_{i = 1}^k {{b_i}{\alpha _i}} \) if and only if there exists no i such that both a_i, b_i are non-zero. In this paper, we show that the graph \(\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)\) is connected and find its domination number, clique number and chromatic number. In the case of finite semirings, we determine the degree of each vertex, order, size, vertex connectivity and girth of \(\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)\). Also, we give a necessary and sufficient condition for \(\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)\) to be Eulerian or Hamiltonian or planar.