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

Authors : Shaohui WANG, Chunxiang WANG, Jia-bao LİU

Pages : 33-40

Doi:10.15672/hujms.519987

View : 14 | Download : 4

Publication Date : 2021-02-04

Article Type : Research Paper

Abstract :For $ \alpha \in [0,1]$, let $A_{\alpha}insert ignore into journalissuearticles values(G); = \alpha Dinsert ignore into journalissuearticles values(G); +insert ignore into journalissuearticles values(1-\alpha);Ainsert ignore into journalissuearticles values(G);$ be $A_{\alpha}$-matrix, where $Ainsert ignore into journalissuearticles values(G);$ is the adjacent matrix and $Dinsert ignore into journalissuearticles values(G);$ is the diagonal matrix of the degrees of a graph $G$. Clearly, $A_{0} insert ignore into journalissuearticles values(G);$ 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.
Keywords : Adjacent matrix, trees, cacti, bounds