Vertex relations of order divisor graphs of subgroups of finite groups

Abstract

Let S(G) = H : H is a subgroup of G. An undirected simple graph Γ(G) is called an order divisor graph of subgroups of a finite group G whose vertex set is S(G) and two distinct vertices H, K ∈ S(G) are adjacent in Γ(G) if and only if either | H | divides | K | or | K | divides | H |. In this paper, we study the
relationships between the vertices of the order divisor graphs of subgroups of finite groups, we show that if H is a subgroup of a finite group G, the degree of H in the order divisor graph of the subgroups of G is greater or
equal to 2. We also establish that there is always a path between two arbitrary vertices of the graph and the vertices of the non triangle-free order divisor graph of S(G) always form atleast {n − (| S(G) | −2)C3} cycles of 3.