The effect on A α -eigenvalues of mixed graphs and unit gain graphs by adding edges in clusters

Abstract

Given a simple graph G, a -cluster of G is a pair of vertex subsets , where size of C is and every vertex in C shares the same set S of s neighbours. Let be a mixed graph whose underlying graph G contains a -cluster and let be a mixed graph on c vertices. Then is a graph obtained from by adding edges between some vertices in C such that . Let and be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. Then the -matrix is defined to be , where . Assume that is a mixed graph containing a cognate -cluster , and for any two mixed graphs on c vertices satisfying each row sum of is equal to that of . We show that and share part of their -eigenvalues. Similarly, we consider the above problem on the unit gain graph. All of these results extend those of Cardoso and Rojo [Edge perturbation on graphs with clusters: adjacency, Laplacian and signless Laplacian eigenvalues. Linear Algebra Appl. 2017;512:113–128.] and those of Belardo, Brunetti and Ciampella [Edge perturbation on signed graphs with clusters: adjacency and Laplacian eigenvalues. Discrete Appl Math. 2019;269:130–138]. Based on our obtained results, we construct some pairs of -cospectral mixed graphs (with respect to two kinds of matrices for mixed graphs) which may not be switching equivalent.