Power graphs: A survey
Jemal H. Abawajy,Andrei V. Kelarev,Morshed U. Chowdhury +2 more
- Vol. 1, Iss: 2, pp 125-147
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This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature.Abstract:
This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.read more
Citations
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Book ChapterDOI
Graphs and Digraphs
TL;DR: For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element, so for any given data element or node in the list structure, the authors can talk in terms of a next element and a previous element.
Journal ArticleDOI
The structure and metric dimension of the power graph of a finite group
TL;DR: This work first shows that P G has a transitive orientation, so it is a perfect graph and its core is a complete graph, and uses the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure of P G .
Journal ArticleDOI
Certain properties of the power graph associated with a finite group
TL;DR: The power graph of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other as mentioned in this paper.
Journal ArticleDOI
Certain properties of the power graph associated with a finite group
TL;DR: In this article, the authors investigated the relation between the structure of directed or undirected graphs and the properties of groups in terms of some properties of the associated graph, and showed that many finite groups such as finite simple groups, symmetric groups and the automorphism groups of sporadic simple groups can be uniquely determined by their power graphs among all finite groups.
Journal ArticleDOI
Automorphism groups of supergraphs of the power graph of a finite group
Asma Hamzeh,Ali Reza Ashrafi +1 more
TL;DR: It is proved that the automorphism group of the main supergraphs and cyclic graphs can be written as a combination of Cartesian and wreath products of some symmetric groups.
References
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Journal ArticleDOI
Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey
TL;DR: A comprehensive and unified analysis of symmetry in a wide variety of Cayley graphs of permutation groups, including the star graph, bubble-sort graph, modified bubble- sort graph, complete-transposition graph, prefix-reversal graph, alternating-group graph, binary and base-b (b ≥ 3) hypercube, cube connected cycles, bisectional graph, folded hypercube and binary orthogonal graph is provided.
Journal ArticleDOI
Undirected power graphs of semigroups
TL;DR: In this article, it was shown that the multiplicative semigroup ℤn and its subgroup Un is complete if and only if n = 1,2,4,p or 2p, where p is a Fermat prime.
Journal ArticleDOI
The power graph of a finite group
Peter J. Cameron,Shamik Ghosh +1 more
TL;DR: It is conjectured that two finite groups with isomorphic power graphs have the same number of elements of each order, and that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.
Book
Graph algebras and automata
TL;DR: This work defines graph algebras and reveals their applicability to automata theory and explores assorted monoids, semigroups, rings, codes, and other algebraic structures to outline theorems and algorithms for finite state automata and grammars.
Journal ArticleDOI
Directed Graphs and Combinatorial Properties of Semigroups
Andrei V. Kelarev,Stephen Quinn +1 more
TL;DR: In this paper, a complete description of all commutative semigroups satisfying three other combinatorial properties defined in terms of directed graphs is given, by a graph we mean a directed graph without loops or multiple edges.