Characterization theorems for the spaces of derivations of evolution algebras associated to graphs

TL;DR

This paper characterizes the derivation spaces of evolution algebras linked to graphs, revealing how graph structure influences derivation properties and including cases with matrices of any rank.

Contribution

It provides a complete description of derivation spaces for evolution algebras associated to graphs based on the twin partition, extending known results to new graph classes.

Findings

Derivation space is zero for graphs without large twin classes.

Descriptions of derivation spaces for various finite graphs.

Includes examples with matrices of any rank.

Abstract

It is well-known that the space of derivations of $n$-dimensional evolution algebras with non-singular matrices is zero. On the other hand, the space of derivations of evolution algebras with matrices of rank $n-1$ has also been completely described in the literature. In this work we provide a complete description of the space of derivations of evolution algebras associated to graphs, depending on the twin partition of the graph. For graphs without twin classes with at least three elements we prove that the space of derivations of the associated evolution algebra is zero. Moreover, we describe the spaces of derivations for evolution algebras associated to the remaining families of finite graphs. It is worth pointing out that our analysis includes examples of finite dimensional evolution algebras with matrices of any rank.