Cyclic intersection graph of subgroups of dihedral groups and its properties.

Abstract

Various characteristics of the algebraic structures can be expressed by associating it to the graphs of different groups. In an intersection graph, each vertex conforms to a set wherein two vertices are connected by an edge if and only if their corresponding sets have a non-empty intersection. For a finite group G, a graph of its subgroups can be represented by the vertices that correspond to the subgroups of G. Based on this factor, we defined the cyclic intersection graph of the subgroups of a group G. The cyclic intersection graph of the subgroups of a group G enclosed all the nontrivial subgroups as its vertices together with two adjacent vertices H and K if and only if H intersection K corresponded to a non-trivial cyclic subgroup. Furthermore, the general presentation of the graph on the dihedral groups was obtained. The cyclic intersection graph of the subgroups on dihedral groups is classified as connected or planar. Additionally, some of the invariants of the the cyclic intersection graph of the subgroups on the dihedral groups are given.