The distance Laplacian and distance signless Laplacian spectrum of the subdivision-vertex join and subdivision-edge join of two regular graphs

Abstract

Let [Formula: see text] be a connected graph with a distance matrix [Formula: see text]. Let [Formula: see text] and [Formula: see text] be, respectively, the distance Laplacian matrix and the distance signless Laplacian matrix of graph [Formula: see text], where [Formula: see text] denotes the diagonal matrix of the vertex transmissions in [Formula: see text]. The eigenvalues of [Formula: see text] and [Formula: see text] constitute the distance Laplacian spectrum and distance signless Laplacian spectrum, respectively. The subdivision graph [Formula: see text] of a graph [Formula: see text] is obtained by inserting a new vertex into every edge of [Formula: see text]. We denote the set of such new vertices by [Formula: see text]. The subdivision-vertex join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. The subdivision-edge join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. In this paper, we determine the distance Laplacian and distance signless Laplacian spectra of subdivision-vertex join and subdivision-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. As an application we exhibit some infinite families of cospectral graphs and find the respective spectra of the Jahangir graph [Formula: see text].