Key Takeaway: The graph, graph, graph, and graph are all singular tricyclic graphs with a theoretical probability of more than half, potentially impacting applications in mathematics, natural sciences, and engineering.

Abstract

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths Ps1, Ps2, Ps3 into a vertex, and three end vertices into a vertex on the cycle Ca1, Ca2, Ca3, respectively, is denoted as γ(a1,a2,a3,s1,s2,s3). Note that δ(a1,a2,a3,s1,s2)=γ(a1,a2,a3,s1,1,s2), ζ(a1,a2,a3,s)=γ(a1,a2,a3,1,1,s), φ(a1,a2,a3)=γ(a1,a2,a3,1,1,1). In this paper, we give the necessity and sufficiency that the γ−graph, δ−graph, ζ−graph and φ−graph are singular and prove that the probability that a randomly given γ−graph, δ−graph, ζ−graph or φ−graph being singular is equal to 325512,165256,4364, 2132, respectively. From our main results, we can conclude that such a γ−graph(δ−graph, ζ−graph, φ−graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.