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arXiv 2008.10430math.CO
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一类联图的$A_α$ -谱

On the $A_α$-spectra of some join graphs

Mainak Basunia, Iswar Mahato, M. Rajesh Kannan

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摘要：

设$G$是一个简单的连通图，$A(G)$是$G$的邻接矩阵。如果$D(G)$是$G$的顶点度的对角矩阵，则对于每个实$α\in [0,1]$，矩阵$A_α(G)$被定义为$$A_α(G) = αD(G) + (1- α) A(G).$$矩阵$A_α(G)$的特征值形成$G$的$A_α$谱。令$G_1 \dot{\vee} G_2$，$G_1 \underline{\vee} G_2$，$G_1 \langle \textrm{v} \rangle G_2$和$G_1 \langle \textrm{e} \rangle G_2$分别表示两个图$G_1$和$G_2$的细分点连接，细分边连接，$R$ -点连接和$R$ -边连接.本文利用正则图$G_1$和任意图$G_2$的$A_α$ -特征值计算了$G_1 \dot{\vee} G_2$，$G_1 \underline{\vee} G_2$，$G_1 \langle \textrm{v} \rangle G_2$和$G_1 \langle \textrm{e} \rangle G_2$的$A_α$ -谱.作为这些结果的应用，我们构造了无穷多对$A_α$ -余谱图.

英文摘要：

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $α\in [0,1]$, the matrix $A_α(G)$ is defined as $$A_α(G) = αD(G) + (1- α) A(G).$$ The eigenvalues of the matrix $A_α(G)$ form the $A_α$-spectrum of $G$. Let $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ denote the subdivision-vertex join, subdivision-edge join, $R$-vertex join and $R$-edge join of two graphs $G_1$ and $G_2$, respectively. In this paper, we compute the $A_α$-spectra of $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ for a regular graph $G_1$ and an arbitrary graph $G_2$ in terms of their $A_α$-eigenvalues. As an application of these results, we construct infinitely many pairs of $A_α$-cospectral graphs.

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