The Singularity of the K4 Homeomorphic Graph

Let G be a finite simple graph and let A(G) be its adjacency matrix. Then, G is singular if A(G) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers ai≥2, i=1,2,…,6. Insert a1−2, a2−2, a3−2, a4−2, a5−2 and a6−2 vertices in the six edges of the complete graph K4, respectively, then the resulting graph is called the K4 homeomorphic graph, denoted by K(a1,a2,a3,a4,a5,a6). In this paper, we give the necessary and sufficient condition for the singularity of K(a1,a2,a3,a4,a5,a6), and we also show that the probability of a K4 homeomorphic graph K(a1,a2,a3,a4,a5,a6) being a singular graph is equal to 193512.