S‐Graphicable Algebras and Specific Graph Families

ABSTRACT This paper presents new developments in the relationship between ‐graphicable algebras and graphs. Several general algebraic properties of ‐graphicable evolution algebras are established, including characterizations of the annihilator, idempotent elements, and evolution subalgebras. It is also shown that ‐graphicable algebras are non‐solvable, and several results concerning their perfectness are provided. In addition, new families of ‐graphicable algebras are introduced, each associated with well‐known graph types, and the structural relationships among these families are analyzed, revealing significant algebraic connections. Finally, an algorithmic method is presented to determine whether a given evolution algebra is ‐graphicable and, if so, to construct its associated graph.