剖分Q-邻接冠图的广义特征多项式及其应用

设正则图G₁和G₂的剖分Q-邻接点冠图G₁□·_QG₂是由Q(G₁)和|V(G₁)|个点不交的G₂的拷贝,通过连接V(G₁)中第i 个顶点的所有邻点与第i个G₂的拷贝的所有点后得到的图; 剖分Q-邻接边冠图G₁□—〓_QG₂是由Q(G₁)和|I(G₁)|个点不交的G₂的拷贝,通过连接 I(G₁)中第 i个顶点的所有邻点与第i个G₂的拷贝的所有点后得到的图。其中Q(G₁)是由图G₁的每条边上插入一个新点且当图G₁的2条边相邻时对应的2个新点之间连接一条边后得到的图, I(G₁)是图G₁中每条边上插入的新点所构成的集合。分别确定了剖分Q-邻接点冠图G₁□·_QG₂和剖分Q-邻接边冠图G₁□—〓_QG₂ 的广义特征多项式及其相应的Φ-谱。得到了G₁□·_QG₂和G₁□—〓_QG₂的规范拉普拉斯谱, 同时也构造了一些Φ-同谱无穷类。

Generalized characteristic polynomial of subdivision Q-neighbourhood corona graphs and its application

The subdivision Q-neighborhood vertex corona G1□·_QG2 of two regular graphs G₁ and G₂ for regular is the graph obtained from vertex disjoint union of Q(G1) and |V(G1)| copies of G₂, and by joining the neighbors of the i^th vertex of |V(G1)| to every vertex in the i^th copy of G₂; the subdivision Q-neighborhood edge corona G1□— _QG2 is the graph obtained from vertex disjoint union of Q(G1) and |I(G1)| copies of G₂, and by joining the neighbors of the i^th vertex of I(G1) to every vertex in the i^th copy of G₂, where Q(G1) is obtained from G₁ by inserting a new vertex into every edge of G₁ and then joining by edges those pair of new vertices which lie on adjacent edges of G₁, and I(G1) is denoted by the set of such new vertices inserted in each edge of G₁. Based on above, the generalized characteristic polynomial and Φ-spectrum of G1□·_QG2 and G1□— _QG2 are determined, respectively. As an application, their normalized Laplacian spectrum are obtained. Besides, infinitely many pairs of Φ-cospectral mates are also constructed.