- Hacettepe Journal of Mathematics and Statistics
- Vol: 50 Issue: 1
- Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices
Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices
Authors : Shaohui Wang, Chunxiang Wang, Jia-bao Liu
Pages : 33-40
Doi:10.15672/hujms.519987
View : 12 | Download : 4
Publication Date : 2021-02-04
Article Type : Research
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.Keywords : Adjacent matrix, trees, cacti, bounds