On some singular graphs with non-isomorphic associated evolution algebras

TL;DR

This paper investigates the isomorphism conditions of evolution algebras associated with graphs, focusing on singular graphs, and provides new examples supporting the conjecture that regularity determines isomorphism.

Contribution

It offers new examples that support the conjecture that regularity or biregularity characterizes isomorphism of associated evolution algebras in singular graphs.

Findings

Confirmed conjecture for certain singular graphs

Provided new examples supporting the conjecture

Extended understanding of evolution algebra isomorphisms

Abstract

A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix of the symmetric random walk on the same graph. It is well-known that, for a non-singular graph, both evolution algebras are isomorphic if, and only if, the graph is regular or biregular. Moreover, through examples and partial results, it has been conjectured that the same result remains true for singular graphs. The purpose of this work is to provide new examples supporting this conjecture.