Distance spectrum of Indu–Bala product of graphs

摘要

The D -eigenvalues μ 1 , μ 2 , … , μ n 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 S p e c D ( G ) . Let G 1 and G 2 be two regular graphs. The Indu–Bala product of G 1 and G 2 is denoted by G 1 ? G 2 and is obtained from two disjoint copies of the join G 1 ∨ G 2 of G 1 and G 2 by joining the corresponding vertices in the two copies of G 2 . In this paper we obtain the distance spectrum of G 1 ? G 2 in terms of the adjacency spectra of G 1 and G 2 . We use this result to obtain a new class of distance equienergetic graphs of diameter 3 . We also prove that the class of graphs K n ˉ ? K n + 1 ˉ has integral distance spectrum.