Singularity transition index of graphs

The adjacency matrix of a simple graph can be singular or non-singular. This paper examines how introducing self-loops alters the diagonal of the adjacency matrix A(G), and hence impacting its determinant and determining whether the graph is singular or non-singular. We define the singularity transition index ST(G), which represents the minimum number of self-loops needed to change the graph’s singular or non-singular nature. Based on ST(G), we classify graphs as labile or inert. For inert graphs, the system A(G_S)X = b has a unique solution for every S and b ∈ ℝⁿ, while for labile graphs which are singular, there exists at least one non-empty set S such that the system has a unique solution for all b, where S consists of vertices with self-loops, and A(G_S) is the corresponding adjacency matrix.