The Singularity of the K4 Homeomorphic Graph - Haicheng Ma (Axioms)
Let G be a finite simple graph and let <semantics>A(G)</semantics> be its adjacency matrix. Then, G is singular if <semantics>A(G)</semantics> is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers <semantics>ai≥2</semantics>, <semantics>i=1,2,…,6</semantics>. Insert <semantics>a1−2</semantics>, <semantics>a2−2</semantics>, <semantics>a3−2</semantics>, <semantics>a4−2</semantics>, <semantics>a5−2</semantics> and <semantics>a6−2</semantics> vertices in the six edges of the complete graph <semantics>K4</semantics>, respectively, then the resulting graph is called the <semantics>K4</semantics> homeomorphic graph, denoted by <semantics>K(a1,a2,a3,a4,a5,a6)</semantics>. In this paper, we give the necessary and sufficient condition for the singularity of <semantics>K(a1,a2,a3,a4,a5,a6)</semantics>, and we also show that the probability of a <semantics>K4</semantics> homeomorphic graph <semantics>K(a1,a2,a3,a4,a5,a6)</semantics> being a singular graph is equal to <semantics><mstyle scriptlevel="0" displaystyle="true">193512</mstyle></semantics>.