Abstract
Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M11) and Section 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.
Highlights
The study of graphical representation of an algebraic structure, especially a semigroup or a group become an energizing research topic over the recent couple of decades, prompting many intriguing outcomes and questions
A natural question is: what will be the effect on connectivity properties of PðGÞ if we remove the identity element from the vertex set of PðGÞ? This section is dedicated to all results based on the connectivity of proper power graph P0ðGÞ of a group G
The Gruenberg–Kegel graph, or prime graph, of a finite group G is the graph whose vertex set is the set of prime divisors of j G j, with an edge joining primes p and q whenever G contains an element of order pq
Summary
The study of graphical representation of an algebraic structure, especially a semigroup or a group become an energizing research topic over the recent couple of decades, prompting many intriguing outcomes and questions In this context, the most well-known class of graphs is the Cayley graph. The concept of directed power graph P~ðGÞ of a group G, introduced by Kelarev and Quinn [51], is a digraph with vertex set G and for any a, b 2 G, there is a directed edge from a to b in P~ðGÞ if and only if ak 1⁄4 b, where k 2 N: For a semi-group, it was first considered in [53] and further studied in [52] All of these papers used the brief term ‘power graph’ to refer to the directed power graph, with the understanding that the undirected power graph is the underlying undirected graph of the directed power graph.
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