关于两个图的一类新连接图的谱

给定2个图G1和G2，设G1的边集E(G1)={e1,e2,?,em1}，则图G1⊙G2可由一个G1，m1个G2通过在G1对应的每条边外加一个孤立点，新增加的点记为U={u1,u2,?,um1}，将ui分别与第i个G2的所有点以及G1中的边ei的端点相连得到，其中i=?1,2,?,m1。得到：（i）当G1是正则图，G2是正则图或完全二部图时，确定了G1⊙G2的邻接谱（A-谱）。（ii）当G1是正则图，G2是任意图时，给出了G1⊙G2的拉普拉斯谱（L-谱）。（iii）当G1和G2都是正则图时，给出了G1⊙G2的无符号拉普拉斯谱（Q-谱）。作为以上结论的应用，构建了无限多对A-同谱图、L-同谱图和Q-同谱图；同时当G1是正则图时，确定了G1⊙G2支撑树的数量和Kirchhoff指数。

On the spectrum of a new join of two graphs

Given graphs G1 and G2,let E(G1)={e1,e2,?,em1} be the edge set of G1,the graph G1⊙G2 can be obtained from one copy of G1 and m1 copies of G2 by adding a new vertex corresponding to each edge of G1, letting the resulting new vertex set be U={u1,u2,?,um1}, and joining ui with each vertex of i-th copy of G2 and with the endpoints of ei, for i=1,2,?,m1. We can determine: (i) the adjacency spectrum of G1⊙G2 for G1,G2 are both regular graphs, or G1 is regular graph, but G2 is a complete bipartite graph; (ii) the Laplacian spectrum of G1⊙G2 when G1 is a regular graph and G2 is an arbitrary graph; (iii) the signless Laplacian spectrum of G1⊙G2 for both G1 and G2 are regular graphs. As applications, we construct infinitely many pairs of A-cospectral graphs, L-cospectral graphs and Q-cospectral graphs. and determine the number of spanning trees and the Kirchhoff index of G1⊙G2,where G1 is a regular graph.