The local partition dimension of graphs

2020 ◽  
pp. 2150028 ◽  
Author(s):  
Ridho Alfarisi ◽  
Arika Indah Kristiana ◽  
Dafik
Keyword(s):  
Sharp Bounds ◽  
Connected Graphs ◽  
Partition Dimension

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.

scholarly journals On Sharp Bounds on Partition Dimension of Convex Polytopes

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 224781-224790
Author(s):  
Yu-Ming Chu ◽  
Muhammad Faisal Nadeem ◽  
Muhammad Azeem ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Convex Polytopes ◽  
Sharp Bounds ◽  
Partition Dimension

scholarly journals Sharp bounds on partition dimension of hexagonal Möbius ladder

2021 ◽  
pp. 101779
Author(s):  
Muhammad Azeem ◽  
Muhammad Imran ◽  
Muhammad Faisal Nadeem
Keyword(s):  
Sharp Bounds ◽  
Partition Dimension

scholarly journals Sharp bounds for partition dimension of generalized Möbius ladders

2018 ◽  
Vol 16 (1) ◽  
pp. 1283-1290 ◽  
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.

scholarly journals The sharp bounds of Zagreb indices on connected graphs

2021 ◽  
pp. 477-489
Author(s):  
Yuming Chu ◽  
Muhammad Kashif Shafiq ◽  
Muhammad Imran ◽  
Muhammad Kamran Siddiqui ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
...  
Keyword(s):  
Sharp Bounds ◽  
Connected Graphs ◽  
Zagreb Indices

Bounds on the partition dimension of convex polytopes

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.

scholarly journals The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

2021 ◽  
Vol 1751 ◽  
pp. 012023
Author(s):  
F C Puri ◽  
Wamiliana ◽  
M Usman ◽  
Amanto ◽  
M Ansori ◽  
...  
Keyword(s):  
Connected Graphs

scholarly journals Sharp Bounds on Davenport-Schinzel Sequences of Every Order

2015 ◽  
Vol 62 (5) ◽  
pp. 1-40 ◽  
Author(s):  
Seth Pettie
Keyword(s):  
Sharp Bounds

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

Sign in / Sign up

Export Citation Format

Share Document