单圈图关联矩阵的特征值

Eigenvalues of Incidence Matrices of Unicyclic Graphs

图的特征值是图的重要指标,目前研究比较多的有图的邻接矩阵特征值,图的拉普拉斯矩阵特征值和图的距离矩阵特征值等等。一般来讲,图的关联矩阵不是方阵因而不存在特征值。图的关联矩阵是方阵当且仅当图是单圈图。在本文中,我们着重于计算单圈图关联矩阵的特征值,证明了其特征值完全反映了圈上的顶点个数和圈外的顶点个数,体现出了特征值能够反应图指标的重要作用。

The eigenvalues of a graph are important indices of this graph,say eigenvalues of its adjacent matrix,eigenvalues of its Laplacian matrix and eigenvalues of its distance matrix.Generally,an incidence matrix of a graph is not square,so it does not have eigenvalues.An incidence matrix of a graph is square if and only if this graph is an unicyclic graph.In this paper,we emphasize on calculating the eigenvalues of the incidence matrices of unicyclic graphs.We show that these eigenvalues clearly reflect the number of vertices and edges both in the cycle and out of the cycle,which once again imply the role of eigenvalues in explaining the graph.