Abstract
In this paper G denotes a finite p-group and Γ denotes a simple undirected graph. The orbit graph is a graph whose vertices are non-central orbit under group action of G on a set. Two vertices v 1 and v 2 are adjacent in the graph if v 1 g = v 2 where v 1 ,v 2 ∈Ω and g∈G. In this paper, the orbit graph of some finite p-groups and group of order pq, where p and q are relatively prime, is found. The orbit graph is determined for the group in the case that a group acts regularly on itself, acts on itself by conjugation, and acts on a set. Besides, some graph properties are found.
Highlights
The concept of graph theory was firstly introduced in 1736 by Leonard Euler who considered Konigsberg bridge problem
The usefulness of graph theory has been proven to a large number of devise fields
The conjugacy class graph was initaly introduced by Bertram[9] in 1990
Summary
The concept of graph theory was firstly introduced in 1736 by Leonard Euler who considered Konigsberg bridge problem. Euler used a graph with vertices and edges to solve this problem. A graph Γ is a mathematical structure consisting of two sets namely vertices V(Γ ) and edges E(Γ ). A subgraph is a graph whose vertices are subset of the vertices and edges of Γ, denoted by Γsub. The complete graph denoted by Kn is a graph whose vertices are adjacent to each other. The followings are some basic concepts related to graph properties that are needed in this article: The independent set[1, 2] is a non-empty set of V(Γ) in which there is no adjacent between two elements of a set in Γ. The clique is the maximum number of complete subgraph, denoted by ω(Γ )[1, 2, 3].
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