符号Wenger图的主特征值

The Main Eigenvalues of Signed Wenger Graphs

摘要: 设是一个符号图，的特征值是其邻接矩阵的特征值。设是的一个特征值，如果存在属于的一个特征向量，使得，则称是的主特征值。图的主特征值为研究图的性质和图的其它不变量都有重要的意义，它在化学等领域中有重要的应用。在的顶点V_i处做切换，相当于改变特征值的特征向量中元素U_i的符号。由于的切换可以改变它的主特征值。因此，给出某些图类的每个特征值都有一无零元素的特征向量具有重要意义。在本文中，我们主要证明了Wenger图的任意特征值对应的特征空间都包含一个无零元素的向量。进一步，我们证明了符号Wenger图在任意地一个顶点处做切换都会使得它的所有特征值变为主特征值。

Abstract: Let be a signed graph and the eigenvalue of be the eigenvalue of its adjacency matrix. An eigenvalue of a signed graph is called a main eigenvalue if it has an eigenvector such that the sum of whose entries is not zero. The main eigenvalue of a graph is of great significance to study the properties of a graph and other invariants of a graph. It has important applications in chemistry and other fields. Switching at the vertex V_i of  is equivalent to changing the sign of the elements U_i in the eigenvector of the eigenvalue . The switching of can change its main eigenvalue. Therefore, it is important to show that every eigenvalue of some graph class has a nowherezero eigenvector. In this paper, we mainly prove that the eigenspace corresponding to any eigenvalue of the Wenger graph contains a nowherezero eigenvector. It is further more prove that all eigenvalues of signed Wenger graphs will become the main eigenvalues when switching at any vertex.