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

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 ${\rho }_{\alpha }(B_{n-d+2,d\mathrm{/}2,d\mathrm{/}2})$ can be computed as the spectral radius of a symmetric tridiagonal matrix of order $\frac{d}{2}+1$ whenever $d$ is even.