On the independence number of the power graph of a finite group

Ma, Xuanlong; Fu, Ruiqin; Lu, Xuefei

doi:10.1016/j.indag.2018.01.002

Cited by 13 publications

(5 citation statements)

References 18 publications

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“…In [79], Tamizh Chelvam et al proved some results on the power graph of a finite abelian group in which they provided a lower bound for the independence number of the power graph of a finite group, computed the independence number of an elementary abelian p-group and characterized all finite abelian groups whose power graph has independence number 2. In 2018, Ma and Lu [59] provided sharp lower and upper bounds for the independence number of PðGÞ and characterized the groups achieving the bounds. Also, they determined the independence number PðGÞ of certain finite groups.…”

Section: Independence Number Of Power Graphsmentioning

confidence: 99%

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Recent developments on the power graph of finite groups – a survey

Kumar

Selvaganesh

Cameron

et al. 2021

AKCE International Journal of Graphs and Combinatorics

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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.

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Section: Independence Number Of Power Graphsmentioning

confidence: 99%

“…[59, Corollary 2.5] Let G be a finite CP group. Then bðPðGÞÞ ¼ X G j j if and only if either G ¼ Z p a or G is a non-cyclic group such that every two maximal cyclic subgroups have trivial intersection.Proposition 7.8.…”

mentioning

confidence: 99%

Recent developments on the power graph of finite groups – a survey

Kumar

Selvaganesh

Cameron

et al. 2021

AKCE International Journal of Graphs and Combinatorics

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“…The chromatic number of power graphs of finite groups is investigated in [12,15] and some results on the independence number of the same is proved in [14].…”

Section: Introductionmentioning

confidence: 99%

Matching in Power Graphs of Finite Groups

2022

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The power graph P(G) of a finite group G is the undirected simple graph with vertex set G, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are investigated. We give upper and lower bounds, and conditions for the power graph of a group to possess a perfect matching. We give a formula for the matching number for any finite nilpotent group. In addition, using some elementary number theory, we show that the matching number of the enhanced power graph $$P_e(G)$$ P e ( G ) of G (in which two elements are adjacent if both are powers of a common element) is equal to that of the power graph of G.

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“…The chromatic number of power graphs of finite groups is investigated in [7] and [9] and some results on the independence number of the same is proved in [8].…”

Section: Introductionmentioning

confidence: 99%