Singularity of Two Kinds of Quadcyclic Peacock Graphs

Let G be a graph. G is singular if and only if the adjacency matrix of graph G is singular. The adjacency matrix of graph G is singular if and only if there is at least one zero eigenvalue. The study of the singularity of graphs is of great significance for better characterizing the properties of graphs. The following definitions are given. There are 4 paths, the starting points of the four paths are bonded into one point, and the ending point of each path is bonded to a cycle respectively, so this graph is called a kind of quadcyclic peacock graph. And in this kind of quadcyclic peacock graph assuming the number of points on the four cycles is a1, a2, a3, a4, and the number of points on the four paths is s1, s2, s3, s4, respectively. This type of graph is denoted by γ (a1, a2, a3, a4, s1, s2, s3, s4), called γ graph. And let γ (a1, a2, a3, a4, 1, 1, 1, 1) = δ (a1, a2, a3, a4), this type four cycles peacock graph called δ graph. In this paper, we give the necessary and sufficient conditions for the singularity of γ graph and δ graph.