Counting Lines and Triangles in the Unit Graphs

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Somnuek Worawiset*

Abstract

Given a group G , define the unit graph gif.latex?\GammaG gif.latex?\Gamma (G,E) to have the vertex-set G and the edge-set E such that for every x,y gif.latex?\in  G,  , X and y  are adjacent in gif.latex?\GammaG if and only if xy = e ,and for each x, y gif.latex?\in G, , and x  and y are adjacent in  gif.latex?\GammaG  where  e is the identity element in a group G. A line in the unit graph is gif.latex?\GammaG  an edge {a,b} gif.latex?\in E such that the degree of is one. A triangle in the unit graph gif.latex?\GammaG  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

Article Details

Section
Original Research Articles

References

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