Counting Lines and Triangles in the Unit Graphs
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Abstract
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
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