On Hilbert evolution algebras of a graph

TL;DR

This paper explores Hilbert evolution algebras linked to graphs, extending finite graph theories to infinite cases, and investigates conditions for their isomorphism, connecting algebraic structures with graph properties.

Contribution

It introduces a new framework for Hilbert evolution algebras associated with graphs, including infinite graphs, and analyzes isomorphism conditions between these structures.

Findings

Defined Hilbert evolution algebras for graphs and symmetric random walks

Extended finite graph theory to infinite graphs

Provided conditions for algebra isomorphism

Abstract

Evolution algebras are a special class of non-associative algebras exhibiting connections with different fields of Mathematics. Hilbert evolution algebras generalize the concept through a framework of Hilbert spaces. This allows to deal with a wide class of infinite-dimensional spaces. In this work we study Hilbert evolution algebras associated to a graph. Inspired in definitions of evolution algebras we define the Hilbert evolution algebra associated to a given graph and the Hilbert evolution algebra associated to the symmetric random walk on a graph. For a given graph, we provide conditions under which these structures are or are not isomorphic. Our definitions and results extend to graphs with infinitely many vertices a similar theory developed for evolution algebras associated to finite graphs.