Monotonic normalized heat diffusion for regular bipartite graphs with four eigenvalues

TL;DR

This paper proves that in bipartite regular graphs with four Laplacian eigenvalues, the normalized heat kernel ratio increases monotonically over time, revealing spectral properties linked to symmetric 2-designs.

Contribution

It establishes a monotonicity property of the heat kernel ratio for a specific class of bipartite graphs with four eigenvalues, connecting spectral graph theory and combinatorial design.

Findings

The heat kernel ratio is monotonically non-decreasing over time.

Graphs with four eigenvalues are incidence graphs of symmetric 2-designs.

The proof leverages the structure of symmetric 2-designs.

Abstract

Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.