Singular difference graphs of vector spaces of square matrices

Abstract: The singular difference graph, denoted by $Γ$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $Γ$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $Γ$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $Γ$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.