Aα spectra of graphs obtained by two corona operations and Aα cospectral graphs

Abstract

Let [Formula: see text] be a graph of which [Formula: see text] be the adjacency matrix and [Formula: see text] be a diagonal matrix whose diagonal entries are the degrees of vertices of [Formula: see text]. [Formula: see text] is known as the signless Laplacian matrix of the graph [Formula: see text]. For any real [Formula: see text] matrix is defined by the convex combination [Formula: see text] of [Formula: see text] and [Formula: see text]. Clearly, [Formula: see text] coincides with [Formula: see text] for [Formula: see text] and coincides with [Formula: see text] for [Formula: see text]. Thus, the [Formula: see text] eigenvalues are generalizations of adjacency eigenvalues and signless Laplacian eigenvalues. In this paper, we have obtained the [Formula: see text] spectra of corona and edge corona of two graphs. We have also shown that [Formula: see text] cospectral graphs and [Formula: see text] equienergetic graphs can be obtained from there.