The Singularity of Three Kinds of New Tricyclic Graphs

A graph G is singular if its adjacency matrix is singular. The starting vertices of two paths Pb1 and Pb2 are simultaneously bound to the ending vertex of the path Ps1, and the ending vertices of the paths Pb1 and Pb2 are bound to the starting vertex of path Ps2. Meanwhile, the starting vertex of the path Ps1 is bound to a vertex of the cycle Ca1, and the ending vertex of the path Ps2 is bound to a vertex of the cycle Ca2. Thus, the resulting graph is written as ξ(a1,a2,b1,b2,s1,s2). This is denoted by ζ(a1,a2,b1,b2,s)=ξ(a1,a2,b1,b2,1,s) and ε(a1,a2,b1,b2)=ζ(a1,a2,b1,b2,1), which are referred to as the ξ-graph, ζ-graph and ε-graph for short, respectively. It is known that there are 15 kinds of tricyclic graphs. The purpose of this paper is to study the necessary and sufficient conditions for ξ-graphs, ζ-graphs and ε-graphs to be singular graphs. We analyzed the structure of the elementary spanning subgraphs of the graph G=ξ(a1,a2,b1,b2,s1,s2). By calculating the determinant of the adjacency matrix of the graph G, the necessary and sufficient conditions for the determinant of the graph G to be zero is obtained, and so the necessary and sufficient conditions for graph ξ(a1,a2,b1,b2,s1,s2) to be singular are obtained. As the corollaries, the necessary and sufficient conditions for graphs ζ(a1,a2,b1,b2,s) and ε(a1,a2,b1,b2) to be singular are also obtained.