Sombor Energy of a Graph with Self-Loops

Abstract

This study aims to extend the notion of degree-based topological index, associated adjacency-type matrix and its energy from a simple graph to a graph with self-loops. Let GS be a graph with k self-loops obtained from a simple graph G, we define Sombor index for GS as SO(GS)where S ⊆ V (G) having self-loop to each of its vertices in S. In addition we investigate some fundamental properties of Sombor eigenvalues, McClelland and Koolen-Moulton-type bound for Sombor energy of GS. Also explores the correlation between Sombor energy of GS and the total π−electron energies associated with the corresponding hetero-molecular systems.

On Spectral Radius and Energy of a Graph with Self-Loops

“…For a real symmetric matrix M with non-zero trace, it is observed that ε ðMÞ: being the sum of absolute values of eigenvalues of M is not equal to E M , i.e., to conclude the definition of ε ðMÞ: given by V. Nikiforov in Equation ( 22), coincides with the graph energies in Equation ( 23) only when the matrix has trace zero. But there are many graph-based matrices studied in literature [2,[12][13][14] which has non-zero traces. Motivated by this, we use a few more properties of matrices and discuss Koolen-Moulton type bound for graph energy in the presence of self-loops.…”