On the $\mathcal{ABS}$ spectrum and energy of graphs

TL;DR

This paper explores the spectral properties of the $ ext{ABS}$ matrix in graphs, characterizing eigenvalues, extremal graphs, and relating $ ext{ABS}$ energy to molecular properties, advancing graph spectral theory and chemical graph analysis.

Contribution

It provides new characterizations of graphs based on $ ext{ABS}$ eigenvalues, bounds for spectral radius and energy, and links to chemical properties, which were not previously established.

Findings

Connected graphs with $ ext{ABS}$ eigenvalue > -1 characterized.

All connected graphs with exactly two $ ext{ABS}$ eigenvalues identified.

Complete bipartite graphs are the only bipartite graphs with three $ ext{ABS}$ eigenvalues.

Abstract

Let $\eta_{1}\ge \eta_{2}\ge\cdots\ge \eta_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $\eta_{n}>-1$. As a result, we determine all connected graphs with exactly two distinct $\mathcal{ABS}$ eigenvalues. We show that a connected bipartite graph has three distinct $\mathcal{ABS}$ eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the $\mathcal{ABS}$ spectral radius (resp. $\mathcal{ABS}$ energy) and characterize extremal graphs. Also, we obtain a relation between $\mathcal{ABC}$ energy and $\mathcal{ABS}$ energy. Finally, the chemical importance of $\mathcal{ABS}$ energy is investigated and it shown that the $\mathcal{ABS}$ energy is useful in predicting certain properties of molecules.