Impact of Self-Loops on the Determinant of Graphs

The current theoretical study intends to analyze the determinant of a certain class of graphs with self-loops. This study focuses on a vast unexplored area of adjacency matrices with nonzero diagonal entities. Further, this study analyzes the implications and properties of the determinants of the adjacency matrices corresponding to a certain class of graphs incorporated with self-loops. Moreover, this study aims to provide a broad insight into the structural and spectral properties of the graph. From the study, it is observed that the determinant of a graph with self-loops GS changes with the number of self-loops and their positions, and so does the singular and nonsingular nature of graph GS. Lastly, this paper aims to examine the nonsingularity of complete, complete bipartite, and some cluster graphs with self-loops by computing all possible determinants. Interestingly, the determinant of a complete graph with self-loop(s) can only have a ternary value (−1, 0, 1), which is noticed and proved in the discussion. Also, the determinant of a complete bipartite graph with self-loops is always nonpositive, as is evident from the study.