Some results on the A_α-eigenvalues of a graph

Abstract

Let G be a simple graph of order n. For $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of G is defined as $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $A\left(G\right)$ and $D\left(G\right)$ are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the $A_{\alpha }$-matrix of G are called the $A_{\alpha }$-eigenvalues of G. In this paper, we first study the properties on $A_{\alpha }$-eigenvalues, i.e. how the $A_{\alpha }$-eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the $A_{\alpha }$-eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively.

Keywords:

• $A_{\alpha }$-eigenvalue
• behaviour
• parameter

AMS Subject Classification:

• 05C50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by NSFC (Nos. 12171089, 12271235), NSF of Fujian (No. 2021J02048).