Singularity Conditions and Probabilities of Tricyclic Graphs

A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) and calculating the determinants of their adjacency matrices, we investigate the necessary and sufficient conditions for the singularity of these three types of tricyclic graphs. Furthermore, we derive that the probabilities of singular graphs among tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) are 217512,113256, and 67128, respectively.