On the structure and spectra of non-inclusion principal ideal graph of inverse semigroups

Abstract

Let S be a semigroup. We define the non-inclusion principal left ideal graph of S as a simple, undirected graph with the nonzero elements of S as vertices and two distinct elements a,b ∈ S are adjacent if and only if a ∉ S¹b and b ∉ S¹a, where S¹a and S¹b are principal left ideals generated by a and b respectively. The non-inclusion principal right ideal graph is defined similarly. Here, we identify the structure of the non-inclusion principal ideal graph of inverse semigroups with a particular emphasis on symmetric inverse semigroups and Brandt semigroups. The energies of some matrices associated with this graph are computed for the Brandt semigroup, yielding a detailed spectral characterization.