On the Structure of the Power Graph and the Enhanced Power Graph of a Group
Copyright policy )On the Structure of the Power Graph and the Enhanced Power Graph of a Group
Let $G$ be a group. The power graph of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph as the enhanced power graph. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.
- Jundi-Shapur University of Technology Iran (Islamic Republic of)
- University of St Andrews United Kingdom
- University of London United Kingdom
- University of Rochester United States
- Rochester Institute of Technology United States
Group Theory (math.GR), 05C25, 05C69, 20D60, clique number, Graphs and abstract algebra (groups, rings, fields, etc.), Coloring of graphs and hypergraphs, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), chromatic number, FOS: Mathematics, Mathematics - Combinatorics, group, Combinatorics (math.CO), independence number, Mathematics - Group Theory, Arithmetic and combinatorial problems involving abstract finite groups
Group Theory (math.GR), 05C25, 05C69, 20D60, clique number, Graphs and abstract algebra (groups, rings, fields, etc.), Coloring of graphs and hypergraphs, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), chromatic number, FOS: Mathematics, Mathematics - Combinatorics, group, Combinatorics (math.CO), independence number, Mathematics - Group Theory, Arithmetic and combinatorial problems involving abstract finite groups
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