Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings

TL;DR

This paper derives an upper bound for the heat kernel of the Laplacian on metric graphs derived from polygonal tilings, capturing both their one-dimensional and two-dimensional characteristics.

Contribution

It provides a novel heat kernel bound for Laplacians on metric graphs from polygonal tilings, linking geometric structure to spectral properties.

Findings

Established an upper heat kernel bound for these graphs.

Demonstrated the bound reflects both 1D and 2D features.

Bridged geometric tiling structures with spectral analysis.

Abstract

We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.