A graph counterexample to davies' conjecture

TL;DR

This paper presents a specific graph example demonstrating that Davies' conjecture, which predicts the convergence of heat kernel ratios, does not hold universally.

Contribution

The authors construct a graph counterexample showing the non-convergence of heat kernel ratios, challenging a previously held conjecture in the field.

Findings

Existence of a graph where heat kernel ratio does not converge

Counterexample to Davies' conjecture

Implications for heat kernel behavior on graphs

Abstract

There exists a graph with two vertices x and y such that the ratio of the heat kernels p(x,x;t)/p(y,y;t) does not converge as t goes to infinity.