Distance spectrum of Indu–Bala product of graphs

Abstract

The D-eigenvalues μ1,μ2,…,μn of a graph G of order n are the eigenvalues of its distance matrix D and form the distance spectrum or D-spectrum of G denoted by SpecD(G). Let G1 and G2 be two regular graphs. The Indu–Bala product of G1 and G2 is denoted by G1▾G2 and is obtained from two disjoint copies of the join G1∨G2 of G1 and G2 by joining the corresponding vertices in the two copies of G2. In this paper we obtain the distance spectrum of G1▾G2 in terms of the adjacency spectra of G1 and G2. We use this result to obtain a new class of distance equienergetic graphs of diameter 3. We also prove that the class of graphs Kn¯▾Kn+1¯ has integral distance spectrum.