Vertex connectivity of the power graph of a finite cyclic group

Chattopadhyay, Sriparna; Patra, Kamal Lochan; Sahoo, Bibhudatta

doi:10.1016/j.dam.2018.06.001

Cited by 19 publications

(23 citation statements)

References 13 publications

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“…The authors of the present paper also independently obtained both the upper bounds (1) and (2) in [9]. Moreover, it was proved that if 2φ(p 1 p 2 · · · p r−1 ) ≥ p 1 p 2 · · · p r−1 , then the bound (1) is sharp, that is, κ(P(C n )) = α(n) [9, Theorem 1.3(i),(iii)].…”

Section: Vertex Connectivitymentioning

confidence: 73%

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

2019

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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.

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Section: Vertex Connectivitymentioning

confidence: 73%

“…If n = p 1 p 2 p 3 is a product of three primes with p 1 < p 2 < p 3 , then κ(P(C n )) ≤ φ(n) + p 1 + p 2 − 1 by [8, Theorem 2.9]. These results were generalized in [9] and [22].…”

Section: Vertex Connectivitymentioning

confidence: 95%

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

2019

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“…One of the major graph representation amongst them is the power graphs of finite groups. We found several papers in this context [2,5,6,9,10,13,14,23,26,27,29]. Example 1.…”

Section: Introductionmentioning

confidence: 99%

Forbidden Subgraphs of Power Graphs

Manna

Cameron

Mehatari

2021

Electron. J. Combin.

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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.

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“…The strong metric dimension of this graph on finite groups was characterized in [15]. In addition, Chattopadhyay et al in [10] determined the exact value for the connectivity of the power graph on finite cyclic groups. Bello et al define order product prime graph of finite groups in [6], as a graph having the elements of a group as its vertices and any two vertices are adjacent if and only if the product of their order is a prime power.…”

Section: Introductionmentioning

confidence: 99%

Graph coloring using commuting order product prime graph

Bello¹,

Ali²,

Isah³

2020

J. Math. Computer Sci.

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The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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Vertex connectivity of the power graph of a finite cyclic group

Cited by 19 publications

References 13 publications

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

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

Forbidden Subgraphs of Power Graphs

Graph coloring using commuting order product prime graph

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