Consider a simple graph G with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G, which is invariant under the symmetry of G. The Sombor index of G is a new graph invariant defined as SO(G)=∑uv∈E(G)(du)2+(dv)2. In this work, we connected the theory of the Sombor index with abstract algebra. We computed this topological index over the tensor and Cartesian products of a monogenic semigroup graph by presenting two different algorithms; the obtained results are illustrated by examples.

Recently, Gutman introduced a class of novel topological invariants named Sombor index which is defined as S O G = ∑ u v ∈ E G d u 2 + d v 2 . In this study, the Sombor index of monogenic semigroup graphs, which is an important class of algebraic structures, is calculated.

In this study, Bruck-Reilly extension of a ternary monoid is defined. Additionally, some results about this construction are given which belongs to one of the classes of ternary semigroups; regular, inverse, orthodox and strongly regular.

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Albertson and the reduced Sombor indices are vertex-degree-based graph invariants that given in [5] and [18], defined as Alb(G)=\sum_{uv\in E(G)}\left|d_{u}-d_{v}\right|, SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{u}-1)^{2}+(d_{v}-1)^{2}}, respectively. In this work we show that a calculation of Albertson and reduced Sombor index which are vertex-degree-based topological indices, over monogenic semigroup graphs.