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.
Categories: math.CO