On the $A_{\alpha}$-characteristic polynomial of a graph

Abstract

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The $A_{\alpha}$-characteristic polynomial of $G$ is defined to be $$ \det(xI_n-A_{\alpha}(G))=\sum_jc_{\alpha j}(G)x^{n-j}, $$ where $\det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_{\alpha}$-spectrum of $G$ consists of all roots of the $A_{\alpha}$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{\alpha}$-spectrum if all graphs having the same $A_{\alpha}$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{\alpha 0}(G)$, $c_{\alpha 1}(G)$, $c_{\alpha 2}(G)$ and $c_{\alpha 3}(G)$ of the $A_{\alpha}$-characteristic polynomial of $G$. And then, we observe that $A_{\alpha}$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_{\alpha}$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_{\alpha}$-spectra.