Maximization of the Spectral Gap for Chemical Graphs by means of a Solution to a Mixed Integer Semidefinite Program

TL;DR

This paper develops a mixed integer semidefinite programming approach to optimize the spectral gap of chemical graphs by bridging two weighted graphs, enhancing understanding of molecular structures through spectral analysis.

Contribution

It introduces a novel mixed integer semidefinite programming method to maximize the spectral gap in chemical graphs, addressing a complex graph construction problem.

Findings

Successfully formulated the spectral gap maximization as a mixed integer semidefinite program

Numerical computations demonstrate the effectiveness of the proposed method

Provides insights into spectral properties relevant for chemical graph analysis

Abstract

In this paper we analyze the spectral gap of a weighted graph which is the difference between the smallest positive and largest negative eigenvalue of its adjacency matrix. Such a graph can represent e.g. a chemical organic molecule. Our goal is to construct a new graph by bridging two given weighted graphs over a bipartite graph. The aim is to maximize the spectral gap with respect to a bridging graph. To this end, we construct a mixed integer semidefinite program for maximization of the spectral gap and compute it numerically.