# Inverse sum indeg energy of graphs

## TL;DR

This paper introduces the inverse sum indeg (ISI) energy of graphs based on the ISI matrix, provides formulas for specific graph classes, and establishes bounds for this new graph energy measure.

## Contribution

It defines the ISI energy of graphs, derives formulas for certain classes, and presents bounds, advancing spectral graph theory with a new energy concept.

## Key findings

- Formulas for ISI energy of some graph classes
- Bounds for ISI energy of graphs
- Introduction of a new spectral graph invariant

## Abstract

Suppose G is an n-vertex simple graph with vertex set {v1,..., vn} and d(i), i = 1,..., n, is the degree of vertex vi in G. The ISI matrix S(G) = [sij] of G is a square matrix of order n and is defined by sij = d(i)d(j)/d(i)+d(j) if the vertices vi and vj are adjacent and sij = 0 otherwise. The S-eigenvalues of G are the eigenvalues of its ISI matrix S(G). Recently the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined as the sum of absolute values of S-eigenvalues of graph G. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs.