Combinatorial Heat and Wave Equations on Certain Classes of Infinite Cayley and Coset Graphs

TL;DR

This paper extends solutions of combinatorial heat and wave equations from finite to infinite Cayley and coset graphs, including non-abelian free groups, revealing solutions as weighted sums over graph balls.

Contribution

It generalizes previous finite graph results to infinite graphs with specific group structures, including non-abelian free groups, and characterizes solutions explicitly.

Findings

Solutions are weighted sums over balls of certain radius.

Extension from finite to infinite graphs with abelian groups.

Explicit solutions on non-abelian free groups.

Abstract

The combinatorial heat and wave equations on all finite Cayley and coset graphs with discrete time variable was solved by Lal {\it et al}. In this paper, the results of the above paper are extended for infinite Cayley and coset graphs, whenever the associated groups are discrete, abelian and finitely generated. Furthermore, we study the solution of the combinatorial heat and wave equations on a $k$-regular tree whose associated group is a non-abelian free group on $k$ generators, each of order~$2$. It turns out that in case of Cayley graphs the solutions to combinatorial heat and wave equations are weighted sum of the initial functions over balls of certain radius which are dependent on the discrete time variable.