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Computing the $A_{\alpha}-$ eigenvalues of a bug. (arXiv:1710.02771v1 [math.CO])

来源于:arXiv
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 查看全文>>