scholarly journals On the Local Adjacency Metric Dimension of Generalized Petersen Graphs

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
Marsidi Marsidi ◽  
Dafik Dafik ◽  
Ika Hesti Agustin ◽  
Ridho Alfarisi
Keyword(s):  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Neighboring Vertex ◽  
Petersen Graphs ◽  
Metric Basis ◽  
Order Set

<p class="Abstract">The local adjacency metric dimension is one of graph topic. Suppose there are three neighboring vertex , ,  in path . Path  is called local if  where each has representation: a is not equals  and  may equals to . Let’s say, .  For an order set of vertices , the adjacency representation of  with respect to  is the ordered -tuple , where  represents the adjacency distance . The distance  defined by 0 if , 1 if  adjacent with , and 2 if  does not adjacent with . The set  is a local adjacency resolving set of  if for every two distinct vertices ,  and  adjacent with y then . A minimum local adjacency resolving set in  is called local adjacency metric basis. The cardinality of vertices in the basis is a local adjacency metric dimension of , denoted by . Next, we investigate the local adjacency metric dimension of generalized petersen graph.</p>

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

scholarly journals Strong Edge Coloring of Generalized Petersen Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1265
Author(s):  
Ming Chen ◽  
Lianying Miao ◽  
Shan Zhou
Keyword(s):  
Edge Coloring ◽  
Petersen Graph ◽  
Chromatic Index ◽  
Induced Matching ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Strong Chromatic Index ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Petersen Graphs

A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n>2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n>2k is at most 9.

scholarly journals A Note on the Distance-Balanced Property of Generalized Petersen Graphs

10.37236/271 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rui Yang ◽  
Xinmin Hou ◽  
Ning Li ◽  
Wei Zhong
Keyword(s):  
Positive Integer ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs ◽  
Balanced Graphs

A graph $G$ is said to be distance-balanced if for any edge $uv$ of $G$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. Let $GP(n,k)$ be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer $k\geq 2$, there exists a positive integer $n_0$ such that the $GP(n,k)$ is not distance-balanced for every integer $n\geq n_0$. In this note, we give a proof of this conjecture.

scholarly journals The Metric Dimension of Some Generalized Petersen Graphs

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Zehui Shao ◽  
S. M. Sheikholeslami ◽  
Pu Wu ◽  
Jia-Biao Liu
Keyword(s):  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs

The distance d(u,v) between two distinct vertices u and v in a graph G is the length of a shortest (u,v)-path in G. For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in G, the code of v with respect to W is the ordered k-tuple cW(v)=(d(v,w1),d(v,w2),…,d(v,wk)). The set W is a resolving set for G if every two vertices of G have distinct codes. The metric dimension of G is the minimum cardinality of a resolving set of G. In this paper, we first extend the results of the metric dimension of P(n,3) and P(n,4) and study bounds on the metric dimension of the families of the generalized Petersen graphs P(2k,k) and P(3k,k). The obtained results mean that these families of graphs have constant metric dimension.

The groups of the generalized Petersen graphs

1971 ◽  
Vol 70 (2) ◽  
pp. 211-218 ◽  
Author(s):  
Roberto Frucht ◽  
Jack E. Graver ◽  
Mark E. Watkins
Keyword(s):  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Trivalent Graph ◽  
Vertex Set ◽  
Petersen Graphs ◽  
Image Position

1.Introduction. For integersnandkwith 2 ≤ 2k <n, thegeneralized Petersen graph G(n, k)has been defined in (8) to have vertex-setand edge-setE(G(n, k))to consist of all edges of the formwhereiis an integer.All subscripts in this paper are to be read modulo n, where the particular value ofnwill be clear from the context. ThusG(n, k)is always a trivalent graph of order 2n, andG(5, 2) is the well known Petersen graph. (The subclass of these graphs withnandkrelatively prime was first considered by Coxeter ((2), p. 417ff.).)

scholarly journals Metric basis and metric dimension of 1-pentagonal carbon nanocone networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Zafar Hussain ◽  
Mobeen Munir ◽  
Ashfaq Ahmad ◽  
Maqbool Chaudhary ◽  
Junaid Alam Khan ◽  
...  
Keyword(s):  
Combinatorial Chemistry ◽  
Connected Graph ◽  
Cardinal Number ◽  
Metric Dimension ◽  
Molecular Topology ◽  
Resolving Set ◽  
Computer Chemistry ◽  
Carbon Nanocones ◽  
Vertex Set ◽  
Metric Basis

AbstractResolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

On the metric dimension of generalized Petersen graphs

2013 ◽  
Vol 36 (3) ◽  
pp. 421-435 ◽  
Author(s):  
Shabbir Ahmad ◽  
Muhammad Anwar Chaudhry ◽  
Imran Javaid ◽  
Muhammad Salman
Keyword(s):  
Metric Dimension ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

scholarly journals Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 300 ◽  
Author(s):  
Zafar Hussain ◽  
Mobeen Munir ◽  
Maqbool Chaudhary ◽  
Shin Kang
Keyword(s):  
Electronic Structure ◽  
Transport Properties ◽  
Present Article ◽  
Connected Graph ◽  
Metric Dimension ◽  
Lattice Structures ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Mathematical Sciences ◽  
Metric Basis

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

Metric Dimension of Generalized Möbius Ladder and its Application to WSN Localization

2020 ◽  
Vol 24 (1) ◽  
pp. 3-11
Author(s):  
Muhammad Idrees ◽  
Hongbin Ma ◽  
Mei Wu ◽  
Abdul Rauf Nizami ◽  
Mobeen Munir ◽  
...  
Keyword(s):  
Global Positioning System ◽  
Wireless Sensor ◽  
Metric Dimension ◽  
Minimal Cardinality ◽  
Resolving Set ◽  
Global Positioning ◽  
Energy Consuming ◽  
Metric Basis ◽  
The Cost ◽  
Key Techniques

Localization is one of the key techniques in wireless sensor network. While the global positioning system (GPS) is one of the most popular positioning technologies, the weakness of high cost and energy consuming makes it difficult to install in every node. In order to reduce the cost and energy consumption only a few nodes, called beacon nodes, are equipped with GPS modules. The remaining nodes obtain their locations through localization. In order to find the minimum positions of beacons, a resolving set with minimal cardinality has been obtained in the network which is called metric basis. Simultaneous local metric basis of the network is also given in which each pair of adjacent vertices of the network is distinguished by some element of simultaneous local metric basis which makes the network design more reasonable. In this paper a new network, the generalized Möbius ladder Mm,n, has been introduced and its metric dimension and simultaneous local metric dimension of its two subfamilies have been calculated.

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