Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Abstract

For $ \alpha \in [0,1]$, let $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ be $A_{\alpha}$-matrix, where $A(G)$ is the adjacent matrix and $D(G)$ is the diagonal matrix of the degrees of a graph $G$. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix. A connected graph is a cactus graph if any two cycles of $G$ have at most one common vertex. We first propose the result for subdivision graphs, and determine the cacti maximizing $A_{\alpha}$-spectral radius subject to fixed pendant vertices. In addition, the corresponding extremal graphs are provided. As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.