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 [Formula: see text], and hence impacting its determinant and determining whether the graph is singular or non-singular. We define the singularity transition index [Formula: see text], which represents the minimum number of self-loops needed to change the graph’s singular or non-singular nature. Based on [Formula: see text], we classify graphs as labile or inert. For inert graphs, the system [Formula: see text] has a unique solution for every [Formula: see text] and [Formula: see text], while for labile graphs which are singular, there exists at least one non-empty set [Formula: see text] such that the system has a unique solution for all [Formula: see text], where [Formula: see text] consists of vertices with self-loops, and [Formula: see text] is the corresponding adjacency matrix.