On Construction of Upper and Lower Bounds for the HOMO-LUMO Spectral Gap

TL;DR

This paper explores how to construct bounds for the HOMO-LUMO spectral gap in graphs formed by bridging two invertible graphs, using semidefinite programming and relaxation techniques to optimize and analyze spectral properties.

Contribution

It introduces a method to bound and optimize the HOMO-LUMO spectral gap in bridged graphs through semidefinite programming and relaxation, providing new tools for spectral graph analysis.

Findings

Derived upper and lower bounds for the HOMO-LUMO gap.

Developed a mixed integer semidefinite programming model for maximization.

Validated methods with computational examples.

Abstract

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.