Eigenvalues of Laplace operators on non-bipartite graphs

TL;DR

This paper analyzes and compares the eigenvalues of Laplace operators under standard and anti-standard conditions on non-bipartite graphs, providing explicit calculations and inequalities for equilateral and inequilateral cases.

Contribution

It introduces a method to compute eigenvalues on equilateral graphs and establishes inequalities between standard and anti-standard eigenvalues on non-bipartite graphs.

Findings

Eigenvalues for equilateral graphs are explicitly calculated.

Inequalities between standard and anti-standard eigenvalues are established.

Comparison methods are extended to inequilateral non-bipartite graphs.

Abstract

This paper considers the comparison between the eigenvalues of Laplace operators with the standard conditions and the anti-standard conditions on non-bipartite graphs which are equilateral or inequilateral. First of all, we show the calculation of the eigenvalues of Laplace operators on equilateral metric graphs with arbitrary edge length. Based on this method, we use the properties of the cosine function and the arccosine function to find the comparison between the eigenvalues of Laplace operators with the standard conditions and the anti-standard conditions on equilateral non-bipartite graphs. In addition, we give the inequalities between standard and anti-standard eigenvalues on a special inequilateral non-bipartite graph.