Abstract
The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$.
 A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.
Highlights
The study of graph representations is one of the interesting and popular research topic in algebraic graph theory
Is a finite group the power graph P (G) is always connected. They further showed that a finite group has a complete undirected power graph if and only if it is cyclic and its order is equal to 1 or pm for some prime p. (We give the proof below.) They counted the number of edges in a power graph of a finite group G by the formula
Our general theme is that various important classes of graphs are defined by forbidden induced subgraphs [4, 31], and we investigate several of these classes with the aim of determining for which groups G the power graph P (G) belongs to the corresponding class
Summary
The study of graph representations is one of the interesting and popular research topic in algebraic graph theory. In [9], the authors proved that if G is a finite group the power graph P (G) is always connected. They further showed that a finite group has a complete undirected power graph if and only if it is cyclic and its order is equal to 1 or pm for some prime p. They showed that if two finite Abelian groups have isomorphic power graphs they are isomorphic They proved that if two finite groups have isomorphic directed power graphs they have same number of elements of each order. The power graph of a finite group G is complete if and only if G is a cyclic group of prime power order. There exist elements of order pn; so G is cyclic
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