On the $A_α$-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 $$ A_α(G)=αD(G)+(1-α)A(G) $$ for any real $α\in [0,1]$. The $A_α$-characteristic polynomial of $G$ is defined to be $$ \det(xI_n-A_α(G))=\sum_jc_{αj}(G)x^{n-j}, $$ where $\det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_α$-spectrum of $G$ consists of all roots of the $A_α$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_α$-spectrum if all graphs having the same $A_α$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{α0}(G)$, $c_{α1}(G)$, $c_{α2}(G)$ and $c_{α3}(G)$ of the $A_α$-characteristic polynomial of $G$. And then, we observe that $A_α$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_α$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_α$-spectra.