Counting Lines and Triangles in the Unit Graphs

Given a group G , define the unit graph _G =  (G,E) to have the vertex-set G and the edge-set E such that for every x,y   G,  , X and y  are adjacent in _G if and only if xy = e ,and for each x, y  G, , and x  and y are adjacent in  _G  where  e is the identity element in a group G. A line in the unit graph is _G  an edge {a,b}  E such that the degree of is one. A triangle in the unit graph _G  is a subgraph which is isomorphic to the cycle of length three. In this paper, we count the number of lines and triangles in the unit graph of some finite groups.

Keywords: group as graphs, graphs of cyclic groups, graphs of dihedral groups, handshaking lemma

^*Corresponding author:

E-mail: wsomnu@kku.ac.th