科研之友
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摘要
Given a simple graph G, a (c, s)-cluster of G is a pair of vertex subsets (C, S), where size of C is c (>= 2) and every vertex in C shares the same set S of s neighbours. Let M-G be a mixed graph whose underlying graph G contains a (c, s)-cluster (C, S) and let M-H be a mixed graph on c vertices. Then M-G(H) is a graph obtained from M-G by adding edges between some vertices in C such that M-G(H)[C] congruent to M-H. Let A(G) and D(G) be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. Then the A(alpha)-matrix is defined to be A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), where alpha is an element of [0, 1]. Assume that MG is a mixed graph containing a cognate (c, s)-cluster (C, S), and for any two mixed graphs M-H, M-H' on c vertices satisfying each row sum of A(alpha)(M-H) is equal to that of A(alpha)(M-H'). We show that M-G(H) and M-G(H') share part of their A(alpha)-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 A(alpha)-cospectral mixed graphs (with respect to two kinds of A(alpha) matrices for mixed graphs) which may not be switching equivalent.
