论文标题
在$a_α$ -spectra上的一些加入图
On the $A_α$-spectra of some join graphs
论文作者
论文摘要
令$ g $为简单，连接的图形，让$ a（g）$为$ g $的邻接矩阵。如果$ d（g）$是$ g $的对角线矩阵，那么对于[0,1] $中的每个实际$α\，矩阵$a_α（g）$被定义为$$a_α（g）=αd（g）=αd（g） +（1-α）a（1-α）a（g）$ a_ $ a_ $ a_ $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ $a_α$ -spectrum $ g $。令$ g_1 \ dot {\ vee} g_2 $，$ g_1 \ undesline {\ vee} g_2 $，$ g_1 \ langle \ textrm {v} \ rangle g_2 $和$ g_1 \ g_1 \ langle \ langle \ langle \ textrm {e} - {e} - {e} \ rangle g_2细分边缘连接，$ r $ -VERTEX JOIN和$ r $ - 两个图的JOIN $ g_1 $和$ g_2 $。在本文中，我们计算$ g_1 \ dot {\ vee} g_2 $，$ g_1 \ useverline {\ vee} g_2 $，$ g_1 \ langle \ langle \ textrm {v} \ rangle g_2 $ g_2 $ g_1 \ langle for for $ g_1 \ dot {\ vee} g_2 $，$ g_1常规图$ g_1 $和任意图$ g_2 $，其$a_α$ - eigenvalues。作为这些结果的应用，我们构建了无限的多对$a_α$ cosectral图。
Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $α\in [0,1]$, the matrix $A_α(G)$ is defined as $$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.
