Abstract
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.
Highlights
We begin with some standard definitions from graph theory and group theory.Let G be a graph with vertex set V (G)
An independent set is a set of vertices in a graph, no two of which are adjacent; that is, a set whose induced subgraph is null
By χ(G), we mean the chromatic number of G, i.e., the minimum number of colours which can be assigned to the vertices of G in such a way that every two adjacent vertices have different colours
Summary
We begin with some standard definitions from graph theory and group theory. Let G be a graph with vertex set V (G). Two finite groups which have isomorphic undirected power graphs have the same number of elements of each order. A new graph pops up while considering these graphs, a graph whose vertex set consists of all group elements, in which two vertices x and y are adjacent if they generate a cyclic group. We call this graph as the enhanced power graph of G and we denote it by Ge(G). For any group G, the clique number of Ge(G) is at most countable
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
