Let G be a finite simple graph and A(G) be its adjacency matrix. Then G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2 , P_a3 is called θ-graph, represented by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively, is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and sufficient conditions for the singularity of α-graph and β-graph, and prove that the probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048 and 733/1024 , respectively.

 Let G be a finite simple graph and A(G) be its adjacency matrix. Then G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2 , P_a3 is called θ-graph, represented by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively, is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and sufficient conditions for the singularity of α-graph and β-graph, and prove that the probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048 and 733/1024 , respectively.