Computing the $A_{\alpha}-$ eigenvalues of a bug

TL;DR

This paper investigates the eigenvalues of the matrix $A_{\alpha}$ for a specific class of graphs called bugs, providing explicit eigenvalue formulas and methods to compute the spectral radius efficiently.

Contribution

It derives explicit eigenvalue formulas for $A_{\alpha}$ of bug graphs and introduces a method to compute the spectral radius using symmetric tridiagonal matrices.

Findings

Eigenvalue $(n-d+2)\alpha -1$ has multiplicity $n-d-1$ in $A_{\alpha}(B)$.

Other eigenvalues, including the spectral radius, are eigenvalues of a symmetric tridiagonal matrix.

Spectral radius of $A_{\alpha}$ for certain bugs can be computed via smaller symmetric tridiagonal matrices.

Abstract

Let $G$ be a simple undirected graph. For $\alpha \in [0,1]$, let \begin{equation*} A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1-\alpha)A\left( G\right) , \end{equation*} where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the degrees of $G$. In particular, $A_{0}(G)=A(G)$ and $A_{\frac{1}{2}}(G)=\frac{1}{2}Q(G)$ where $Q(G)$ is the signless Laplacian matrix of $G$. A bug $B_{p,q,r}$ is a graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. In \cite{HaSt08}, Hansen and Stevanovi\'{c} proved that, among the graphs $G$ of order $n$ and diameter $d$, the largest spectral radius of $A(G)$ is attained by the bug $B_{n-d+2,\lfloor d/2\rfloor,\lceil d/2\rceil}$. In \cite{LiLu14}, Liu and Lu proved the same result for the spectral radius of $Q(G)$. Let $\rho_{\alpha}(G)$ be the spectral radius of $A_{\alpha}(G)$. In this note, for a bug $B$ of order $n$ and diameter $d$, it is shown that $(n-d+2)\alpha -1$ is an eigenvalue of $A_{\alpha}(B)$ with multiplicity $n-d-1$ and that the other eigenvalues, among them $\rho_{\alpha}(B)$, can be computed as the eigenvalues of a symmetric tridiagonal matrix of order $d+1$. It is also shown that $\rho_{\alpha}(B_{n-d+2,d/2,d/2})$ can be computed as the spectral radius of a symmetric tridiagonal matrix of order $\frac{d}{2}+1$ whenever $d$ is even.