Graphs with Clusters Perturbed by Regular Graphs—A_α-Spectrum and Applications

Given a graph G, its adjacency matrix A(G) and its diagonal matrix of vertex degrees D(G), consider the matrix A_α(G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1). The Aα -spectrum of G is the multiset of eigenvalues of A_α(G) and these eigenvalues are the α-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_α(G) and A_α(G(H)) are deduced and if A_α(H) is positive semidefinite, then the i-th eigenvalue of A_α(G(H)) is greater than or equal to i-th eigenvalue of A_α(G). These results are extended to graphs with several pairwise disjoint clusters (C₁, S₁), . . ., (C_k, S_k). As an application, the effect on the energy, α-Estrada index and α-index of a graph G with clusters when the edges of regular graphs are added to G are analyzed. Finally, the A_α-spectrum of the corona product G ◦ H of a connected graph G and a regular graph H is determined.