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

TL;DR

This paper introduces the $A_{\alpha}$-characteristic polynomial of a graph, formulates its coefficients, and demonstrates its effectiveness in distinguishing graphs, especially for graphs with up to 10 vertices.

Contribution

The paper derives explicit formulas for the first four coefficients of the $A_{\alpha}$-characteristic polynomial and shows its utility in graph identification.

Findings

$A_{\alpha}$-spectra effectively distinguish graphs.

Formulas for the first four coefficients of the polynomial.

Identification of some graphs determined by their $A_{\alpha}$-spectra.

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.