The structure and metric dimension of the power graph of a finite group

Feng, Min; Ma, Xuanlong; Wang, Kaishun

doi:10.1016/j.ejc.2014.08.019

Cited by 65 publications

(34 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was proved in [11,Theorem 1.3] and [12,Corollary 3.4] that, among all finite groups of a given order, the cyclic group of that order has the maximum number of edges and has the largest clique in its power graph. By [13,Theorem 5] and [15,Corollary 2.5], the power graph of a finite group is perfect, in particular, the clique number and the chromatic number coincide. Explicit formula for the clique number of the power graph of a finite cyclic group is given in [20,Theorem 2] and [13,Theorem 7].…”

Section: Power Graphmentioning

confidence: 99%

Vertex connectivity of the power graph of a finite cyclic group II

2019

View full text Add to dashboard Cite

The power graph P(G) of a given finite group G is the simple undirected graph whose vertices are the elements of G, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity κ(P(G)) of P(G) is the minimum number of vertices which need to be removed from G so that the induced subgraph of P(G) on the remaining vertices is disconnected or has only one vertex. For a positive integer n, let C n be the cyclic group of order n. Suppose that the prime power decomposition of n is given by n = p n 1 1 p n 2 2 · · · p nr r , where r ≥ 1, n 1 , n 2 , . . . , n r are positive integers and p 1 , p 2 , . . . , p r are prime numbers with p 1 < p 2 < · · · < p r . The vertex connectivity κ(P(C n )) of P(C n ) is known for r ≤ 3, see [22,9]. In this paper, for r ≥ 4, we give a new upper bound for κ(P(C n )) and determine κ(P(C n )) when n r ≥ 2. We also determine κ(P(C n )) when n is a product of distinct prime numbers.

show abstract

Section: Power Graphmentioning

confidence: 99%

Vertex connectivity of the power graph of a finite cyclic group II

2019

View full text Add to dashboard Cite

show abstract

“…Actually, given a graph Γ, determining whether Γ is power-critical is equivalent to determining whether Γ is isomorphic to a spanning subgraph of the power graph of some finite group. Many researchers [6,9,19] investigated two groups which have isomorphic power graphs. Actually, this problem is to find two finite groups G and H such that G is Γ H -optimal and H is Γ G -optimal.…”

Section: Definition 12mentioning

confidence: 99%

The Power Index of a Graph

Feng

Wang

2017

Graphs and Combinatorics

Self Cite

View full text Add to dashboard Cite

The power index Θ(Γ) of a graph Γ is the least order of a group G such that Γ can embed into the power graph of G. Furthermore, this group G is Γ-optimal if G has order Θ(Γ). We say that Γ is power-critical if its order equals to Θ(Γ). This paper focuses on the power indices of complete graphs, complete bipartite graphs and 1-factors. We classify all power-critical graphs Γ in these three families, and give a necessary and sufficient condition for Γ -optimal groups.

show abstract

“…Later in [6] Cameron proved that finite groups with isomorphic power graphs have isomorphic directed power graphs. The authors of [1] and [9] independently proved that the power graph of any group of bounded exponent is perfect. In [12] Shitov proved that chromatic number of the power graph of any semigroup is at most countable.…”

Section: Introductionmentioning

confidence: 99%