Abstract

Given a graph G, its adjacency matrix A(G) and its diagonal matrix of vertex degrees D(G), consider the matrix A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), where alpha is an element of [0, 1). The Aff-spectrum of G is the multiset of eigenvalues of A(alpha)(G) and these eigenvalues are the alpha-eigenvalues of G. A cluster in G is a pair of vertex subsets (C, S), where C is a set of cardinality |C | >= 2 of pairwise co-neighbor vertices sharing the same set S of |S | neighbors. Assuming that G is connected and it has a cluster (C, S), G(H) is obtained from G and an r-regular graph H of order |C | by identifying its vertices with the vertices in C, eigenvalues of A(alpha)(G) and A(alpha)(G(H)) are deduced and if A(alpha)(H) is positive semidefinite, then the i-th eigenvalue of A(alpha)(G(H)) is greater than or equal to i-th eigenvalue of A(alpha)(G). These results are extended to graphs with several pairwise disjoint clusters (C-1, S-1),..., (C-k, S-k). As an application, the effect on the energy, alpha-Estrada index and ff-index of a graph G with clusters when the edges of regular graphs are added to G are analyzed. Finally, the A(alpha)-spectrum of the corona product G omicron H of a connected graph G and a regular graph H is determined.