Prime, composite and fundamental Kirchhoff graphs

TL;DR

This paper introduces an algorithm to construct all Kirchhoff graphs up to a certain edge multiplicity, explores prime Kirchhoff graph tilings, and proves the existence of infinitely many such graphs with a minimal multiplicity.

Contribution

It presents a novel algorithm for constructing Kirchhoff graphs and establishes key properties about prime Kirchhoff graphs and their minimal multiplicity.

Findings

Algorithm constructs all Kirchhoff graphs up to fixed edge multiplicity

Infinitely many prime Kirchhoff graphs exist for given fundamental graphs

Minimal multiplicity for Kirchhoff graph existence is established

Abstract

A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. An algorithm has been developed that constructs all the Kirchhoff graphs up to a fixed edge multiplicity. This algorithm is used to explore the structure of prime Kirchhoff graph tilings. The existence of infinitely many prime Kirchhoff graphs given a set of fundamental Kirchhoff graphs is established, as is the existence of a minimal multiplicity for Kirchhoff graphs to exist.