Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs

Abstract

Recently Indulal and Balakrishnan [Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13 (2016) 230–234] put forward a new graph operation, namely, the Indu–Bala product [Formula: see text] of graphs [Formula: see text] and [Formula: see text], and it is obtained from two disjoint copies of the join [Formula: see text] of [Formula: see text] and [Formula: see text] by joining the corresponding vertices in the two copies of [Formula: see text]. In this paper, we obtain the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of [Formula: see text] in terms of the corresponding spectra of [Formula: see text] and [Formula: see text]. As applications, these results enable us to construct infinitely many pairs of respective cospectral graphs. Further, the Laplacian spectra enable us to get the formulas of the number of spanning trees and Kirchhoff index of [Formula: see text] in terms of the Laplacian spectra of regular graphs [Formula: see text] and [Formula: see text].