On the Vertex Seidel Energy of Graphs

TL;DR

This paper introduces the vertex Seidel energy, a new graph invariant based on the Seidel matrix, providing spectral formulas, exact values for specific graphs, bounds, and invariance properties.

Contribution

The paper defines the vertex Seidel energy, derives its spectral formula, computes exact values for certain graphs, and proves its invariance under Seidel switching and graph complementation.

Findings

Vertex Seidel energy is invariant under Seidel switching.

Exact values computed for several graph families.

Bounds and integral representations established.

Abstract

We introduce the vertex Seidel energy via the diagonal entries of the absolute Seidel matrix. We establish a spectral formula, compute exact values for several graph families, derive bounds, and present a Coulson-type integral representation for analytical study of this invariant. We also show that vertex Seidel energy is invariant under Seidel switching and complementation.