$A_{\alpha }$-spectrum of a graph obtained by copies of a rooted graph and applications

Abstract

Given a connected graph $R$ on $r$ vertices and a rooted graph $H,$ let $R\{H\}$ be the graph obtained from $r$ copies of $H$ and the graph $R$ by identifying the root of the $i-th$ copy of $H$ with the $i-th$ vertex of $R$. Let $0\le \alpha \le 1,$ and let \[ A_α(G)=αD(G)+(1-α)A(G) \] where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. A basic result on the $A_{\alpha }-$ spectrum of $R\{H\}$ is obtained. This result is used to prove that if $H=B_k$ is a generalized Bethe tree on $k$ levels, then the eigenvalues of $A_{\alpha }(R\{B_k\})$ are the eigenvalues of symmetric tridiagonal matrices of order not exceeding $k$; additionally, the multiplicity of each eigenvalue is determined. Finally, applications to a unicyclic graph are given, including an upper bound on the $\alpha -$ spectral radius in terms of the largest vertex degree and the largest height of the trees obtained by removing the edges of the unique cycle in the graph.