Connecting boundary and interior - "Gauss's law" for graphs

TL;DR

This paper explores a graph-theoretic analogue of Gauss's law, establishing a boundary-interior relation by defining a flux concept for graphs, extending classical physical principles to discrete structures.

Contribution

It introduces a novel definition of flux for graphs that satisfies a Gauss's law-like boundary-interior relation, bridging concepts from physics and graph theory.

Findings

Established a boundary-interior relation for graphs

Proposed a new flux definition for graph structures

Extended physical laws to discrete mathematical objects

Abstract

The Gauss's law, in an abstract sense, is a theorem that relates quantities on the boundary (flux) to the interior (charge) of a surface. An identity for soap froths were proved with the same boundary-interior relation. In this article, we try to construct a definition of flux for other graphs, such that a similar boundary-interior relation can be satisfied.