Sharp bounds on partition dimension of hexagonal Möbius ladder

All graphs in this paper are undirected and connected graphs. An ordered k-partition set [Formula: see text] where [Formula: see text], the representation of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text] where [Formula: see text] is the distance between the vertex v and the set [Formula: see text] with [Formula: see text] for [Formula: see text]. The partition set [Formula: see text] is a local resolving partition of [Formula: see text] if [Formula: see text] for [Formula: see text] adjacent to [Formula: see text] of [Formula: see text]. The minimum local resolving partition [Formula: see text] is a local partition dimension of [Formula: see text], denoted by [Formula: see text]. In our paper, we found the sharp bounds of the local partition dimension of [Formula: see text] and determine the exact value of some special graph.

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Sharp bounds for partition dimension of generalized Möbius ladders

Open Mathematics ◽

10.1515/math-2018-0109 ◽

2018 ◽

Vol 16 (1) ◽

pp. 1283-1290 ◽

Cited By ~ 2

Author(s):

Zafar Hussain ◽

Junaid Alam Khan ◽

Mobeen Munir ◽

Muhammad Shoaib Saleem ◽

Zaffar Iqbal

Keyword(s):

Lower Bounds ◽

Metric Space ◽

Robot Navigation ◽

Upper Bounds ◽

Pivotal Role ◽

Resolving Set ◽

Sharp Bounds ◽

Partition Dimension

AbstractThe concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

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Sharp Bounds on Treatment Effects

SSRN Electronic Journal ◽

10.2139/ssrn.2197073 ◽

2013 ◽

Cited By ~ 1

Author(s):

Ismael Mourifie

Keyword(s):

Treatment Effects ◽

Sharp Bounds

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Bounds on the partition dimension of convex polytopes

Combinatorial Chemistry & High Throughput Screening ◽

10.2174/1386207323666201204144422 ◽

2020 ◽

Vol 23 ◽

Author(s):

Jia-Bao Liu ◽

Muhammad Faisal Nadeem ◽

Mohammad Azeem

Keyword(s):

Combinatorial Optimization ◽

Computer Science ◽

Facility Location ◽

Robot Navigation ◽

Convex Polytopes ◽

Pharmaceutical Chemistry ◽

Resolving Set ◽

Minimum Number ◽

Partition Dimension ◽

Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

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Sharp Bounds on Davenport-Schinzel Sequences of Every Order

Journal of the ACM ◽

10.1145/2794075 ◽

2015 ◽

Vol 62 (5) ◽

pp. 1-40 ◽

Cited By ~ 3

Author(s):

Seth Pettie

Keyword(s):

Sharp Bounds

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The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

Mathematica Slovaca ◽

10.1515/ms-2017-0398 ◽

2020 ◽

Vol 70 (4) ◽

pp. 849-862

Author(s):

Shagun Banga ◽

S. Sivaprasad Kumar

Keyword(s):

Triangle Inequality ◽

Starlike Functions ◽

Sharp Bound ◽

The Novel ◽

Sharp Bounds ◽

Hankel Determinants ◽

Lemniscate Of Bernoulli ◽

Carathéodory Functions ◽

The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

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Coefficient inequalities for a subclass of Bazilevič functions

Demonstratio Mathematica ◽

10.1515/dema-2020-0040 ◽

2020 ◽

Vol 53 (1) ◽

pp. 27-37

Author(s):

Sa’adatul Fitri ◽

Derek K. Thomas ◽

Ratno Bagus Edy Wibowo ◽

Keyword(s):

Recent Work ◽

Sharp Bounds ◽

Coefficient Problems ◽

Coefficient Inequalities ◽

Bazilevic Functions

AbstractLet f be analytic in {\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}, and for α ≥ 0 and 0 < λ ≤ 1, let { {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying \left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f\in { {\mathcal B} }_{1}(\alpha ,\lambda ), thus extending recent work in the case λ = 1.

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The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/s10231-021-01089-3 ◽

2021 ◽

Author(s):

Young Jae Sim ◽

Adam Lecko ◽

Derek K. Thomas

Keyword(s):

Unit Disk ◽

Convex Functions ◽

Univalent Functions ◽

Inverse Function ◽

Invariance Property ◽

Hankel Determinant ◽

Sharp Bounds ◽

Strongly Convex Functions ◽

Strongly Convex ◽

Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

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Bounds of Fractional Metric Dimension and Applications with Grid-Related Networks

Mathematics ◽

10.3390/math9121383 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1383

Author(s):

Ali H. Alkhaldi ◽

Muhammad Kamran Aslam ◽

Muhammad Javaid ◽

Abdulaziz Mohammed Alanazi

Keyword(s):

Metric Dimension ◽

Hard Problem ◽

Np Hard ◽

Weighted Version ◽

Sharp Bounds ◽

Np Hard Problem ◽

Metric Dimensions

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional metric dimension for all the connected networks is an NP-hard problem. In this note, we find the sharp bounds of the fractional metric dimensions of all the connected networks under certain conditions. Moreover, we have calculated the fractional metric dimension of grid-like networks, called triangular and polaroid grids, with the aid of the aforementioned criteria. Moreover, we analyse the bounded and unboundedness of the fractional metric dimensions of the aforesaid networks with the help of 2D as well as 3D plots.

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