On Hybrid Order-Sum Graphs of Finite Dihedral Groups

Abstract

This paper introduces a novel graph construction, the inverse-order sum graph, for the dihedral group D₂ₙ. By merging the adjacency conditions of the inverse graph and the order sum graph, we define Γᵢᵥₒₛ(D₂ₙ) and investigate its fundamental graph properties. For n odd, we establish explicit formulas for vertex degrees, graph sizes, and completeness. The inverse graph Γ_Iv (D₂ₙ) exhibits the highest connectivity with degrees n-1 in P₁ ∪ P₃ and n-2 in P₂, while the order sum graph Γ_OS (D₂ₙ) is sparser with edges only between full-order rotations. The inverse-order sum graph Γ_IvOS (D₂ₙ) is the most restrictive, yielding n-3 degrees in P₃ and isolated vertices elsewhere. Our comparative analysis reveals strict inclusion relations and structural hierarchies among these graphs, demonstrating how combining algebraic conditions produces refined graphical representations of group elements. These results contribute to algebraic graph theory by providing new tools for analyzing finite group structures through hybrid graph constructions, with potential applications in group-based cryptography and network modeling.