A Descent on Simple Graphs -- from Complete to Cycle -- and Algebraic Properties of Their Spectra

TL;DR

This paper explores the spectral properties of simple graphs during a stepwise edge removal process from complete to cycle graphs, revealing that large-diameter graphs tend to have eigenvalues of high algebraic degree.

Contribution

It introduces a novel graph descent method and establishes a link between graph diameter and the algebraic degree of eigenvalues.

Findings

Graphs with large diameter have eigenvalues of high algebraic degree.

Quantitative analysis of spectral changes during graph descent.

Spectral properties are closely related to graph structure and edge removal sequence.

Abstract

We investigate a descent on simple graphs, starting with the complete graph on $n$ vertices and ending up with the cycle graph by removing one edge after another. We obtain quantitative results showing that graphs with large diameter must have some eigenvalues of large algebraic degree.