Power graphs of (non)orientable genus two

Ma, Xuanlong; Walls, Gary L.; Wang, Kaishun

doi:10.1080/00927872.2018.1476522

Cited by 25 publications

(5 citation statements)

References 30 publications

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“…Further, Doostabadi et al [11], characterized the finite groups whose power graphs are of (non)orientable genus one. Then all the finite groups with (non)orientable genus two power graphs have been characterized in [28]. The undirected power graphs of groups have been studied in other aspects, see [7,9,13,24,26] and references therein.…”

Section: Historical Background and Main Resultsmentioning

confidence: 99%

Nilpotent groups whose Difference graphs have positive genus

Parveen¹

2023

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The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is a simple undirected graph whose vertex set is the group G and two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we study the difference graph D(G) of a finite group G which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We characterize all the finite nilpotent groups G such that the genus (or cross-cap) of the difference graph D(G) is at most 2. 2020 Mathematics Subject Classification. 05C25. Key words and phrases. Enhanced power graph, power graph, nilpotent groups, genus and cross-cap of a graph.

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Section: Historical Background and Main Resultsmentioning

confidence: 99%

Nilpotent groups whose Difference graphs have positive genus

Parveen¹

2023

Preprint

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“…The concepts of power graph and undirected power graph were first introduced by Kelarev and Quinn [18] and Chakrabarty et al [6], respectively. In recent years, the study of power graphs has been growing, see, for example, [4,5,21,22]. Also, see [1] for a survey of results and open problems on power graphs.…”

Section: Introductionmentioning

confidence: 99%

On the Metric Dimension of the Reduced Power Graph of a Finite Group

2022

Taiwanese J. Math.

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Let G be a finite group. The reduced power graph of G is the undirected graph whose vertex set is G, and two distinct vertices x and y are adjacent if x ⊂ y or y ⊂ x . In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a P-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

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“…This article only considers the undirected power graphs corresponding to finite groups. Later on, in [2,3] it was shown that the power graph of two finite groups G and H with the same order, are isomorphic if and only if given any d | |G|, the number of elements of G of order d is equal to the number of elements of H of order d. The strong metric dimension of the power graph of a finite group is studied in [9] and the power graphs which are non-orientable and genus two are studied in [8]. For a detailed survey on the power graphs and related areas, see the survey articles [1], and [6].…”

Section: Introductionmentioning

confidence: 99%

The Lambda Number of the Power Graph of Finite Simple Groups

Sarkar¹

2022

Preprint

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For a finite group G, the power graph ΓG is a graph whose set of vertices is equal to G and two distinct elements of G are adjacent if and only if one of them is a positive integer power of the other. An L(2, 1)-labelling of ΓG is an integer-valued function defined on G so that the distance between the images of two adjacent vertices (resp. vertices with distance two) are at least two (resp. at least one). The lambda number λ(G) is defined to be the least difference between the largest and the smallest integer values assigned to the vertices of ΓG for all possible L(2, 1)-labellings of ΓG. It is known that λ(G) ≥ |G|. In this paper, we prove that if G is a finite simple group, then λ(G) = |G| except when G is cyclic of prime order. This settles a partial classification of finite groups G that achieve the lower bound on of lambda number.

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Power graphs of (non)orientable genus two

Cited by 25 publications

References 30 publications

Nilpotent groups whose Difference graphs have positive genus

Nilpotent groups whose Difference graphs have positive genus

On the Metric Dimension of the Reduced Power Graph of a Finite Group

The Lambda Number of the Power Graph of Finite Simple Groups

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