Graphs with clusters perturbed by regular graphs: Aα-spectrum and applications

Abstract

Given a graph $G$, its adjacency matrix $A(G)$ and its diagonal matrix of vertex degrees $D(G)$, consider the matrix $A_{\alpha}\left( G\right) = \alpha D\left( G\right) +(1-\alpha)A\left(G\right)$, where $\alpha \in\left[ 0,1\right)$. The $A_{\alpha}-$ 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 $|C| \ge 2$ 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 $(C_1,S_1), \ldots, (C_k,S_k)$. As an application, the effect on the energy, $\alpha$-Estrada index and $\alpha$-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 $G \circ H$ of a connected graph $G$ and a regular graph $H$ is determined.