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 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 Metric dimension and pattern avoidance in graphs

2018 ◽  
Author(s):  
Jesse Geneson
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Pattern Avoidance ◽  
Metric Dimension ◽  
Maximum Value ◽  
Vertex Graph ◽  
Dimension 2

We prove that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$ is at most $(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i-1)^{k-1}$, sharpening the bound of $\binom{k}{2}+k D^{k-1}+D^{k}$ from Zubrilina (2018). We also prove that there is no subgraph of diameter $D$ with more than $(D+1)^{k}$ vertices in any connected graph of metric dimension $k$, and there is no subgraph of diameter $D$ with more than $(D+1)^{k}$ edges in any connected graph of edge metric dimension $k$. In particular, we prove that the maximum value of $n$ such that a graph of metric dimension $\leq k$ can contain the complete graph $K_{n}$ as a subgraph is $n = 2^{k}$. We also prove that the maximum value of $n$ such that a graph of metric dimension or edge metric dimension $\leq k$ can contain the complete bipartite graph $K_{n,n}$ as a subgraph is $2^{\theta(k)}$. Furthermore, we show that the maximum value of $n$ such that a graph of edge metric dimension $\leq k$ can contain $K_{1,n}$ as a subgraph is $n = 2^{k}$. We also show that the maximum value of $n$ such that a graph of metric dimension $\leq k$ can contain $K_{1,n}$ as a subgraph is $3^{k}-O(k)$. In addition, we prove that the $d$-dimensional grid $\prod_{i = 1}^{d} P_{r_{i}}$ has edge metric dimension at most $d$. This generalizes two results of Kelenc et al (2016), that non-path grids have edge metric dimension $2$ and that $d$-dimensional hypercubes have edge metric dimension at most $d$. We also provide a characterization of $n$-vertex graphs with edge metric dimension $n-2$, answering a question of Zubrilina. As a result of this characterization, we prove that any connected $n$-vertex graph $G$ such that $edim(G) = n-2$ has diameter at most $5$. More generally, we prove that any connected $n$-vertex graph with edge metric dimension $n-k$ has diameter at most $3k-1$.

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 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.

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Hussain
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Four Dimensions ◽  
Symmetric Graphs ◽  
Metric Dimensions ◽  
Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

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.

scholarly journals Resolvability in Subdivision of Circulant Networks Cn1,k

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Jianxin Wei ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Ghulam Abbas ◽  
Muhammad Imran
Keyword(s):  
Facility Location ◽  
Connected Graph ◽  
Location Problems ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Facility Location Problems ◽  
Wide Range ◽  
Circulant Networks

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

scholarly journals Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Strong Metric Dimension ◽  
Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

scholarly journals Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

2021 ◽  
Vol 9 ◽  
Author(s):  
Sunny Kumar Sharma ◽  
Hassan Raza ◽  
Vijay Kumar Bhat
Keyword(s):  
Combinatorial Chemistry ◽  
Connected Graph ◽  
Metric Dimension ◽  
Molecular Topology ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Crucial Information ◽  
Metric Basis

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.

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