Mesure et action des i-permutations sur les multigraphes multicolores finis et inifinis

TL;DR

This paper introduces a vector space framework for multicolored multigraphs, demonstrating a method to characterize graph isomorphism through measures of subgraph counts, applicable to finite and infinite graphs.

Contribution

It establishes an $ ext{R}$-vector space with a basis for representing graphs and links graph isomorphism to measure-based subgraph counts, offering a novel algebraic approach.

Findings

Existence of an $ ext{R}$-vector space basis for graphs.

Representation of graphs as linear combinations of basis elements.

Graph isomorphism characterized by measure-preserving subgraph counts.

Abstract

Among other results, the purpose of this article is to show the existence of an $\mathbb{R}$-space-vector with basis $\omega^i_j $, $i, j$ are integers such that every graph with n vertex $n \geq 3$ is the vector: $$\mathcal{V}(n) = \sum_{j=0}^{n-1} {\alpha^{n-1}_j} \omega^{n-1}_j $$ where $\alpha^{n-1}_j$ is the number of sub graphs of type $\omega^{n-1}_j$ . We deduce that two graphs are isomorphic if for any measure, they have the same number of maximal proper subset with this measure.