scholarly journals Pelabelan 0-Anti Ajaib dan 2-Anti Ajaib untuk Graf Tangga L

2016 ◽  
Vol 12 (1) ◽  
pp. 63
Author(s):  
Quinoza Guvil ◽  
Roni Tri Putra
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Partition Dimension ◽  
Dimension Partition

For a connected graph  and a subset  of   . For a vertex  the distance betwen  and  is . For an ordered k-partition of ,  the representation of   with respect to  is    The k-partition  is a resolving partition if  are distinct for every  The minimum k for which there is a resolving partition of   is the partition dimension of   In this paper will shown resolving partition of  connected graph order  where  is a bipartite graph. Then it is shown dimension partition of bipartite graph, are pd(Kst)=n-1

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].

THE SUPER SPANNING CONNECTIVITY AND SUPER SPANNING LACEABILITY OF TORI WITH FAULTY ELEMENTS

2013 ◽  
Vol 24 (06) ◽  
pp. 921-939 ◽  
Author(s):  
JING LI ◽  
DI LIU ◽  
JUN YUAN
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Disjoint Paths ◽  
Two Dimensional ◽  
Dimensional Torus

A k-container of a graph G is a set of k internally disjoint paths between two distinct vertices. A k-container of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices, and a bipartite graph G is k*-laceable if there exists a k*-container between any two vertices from different partite sets of G. A k-connected graph (respectively, bipartite graph) G is super spanning connected (respectively, laceable) if G is r*-connected (r*-laceable) for any r with 1 ≤ r ≤ k. This paper shows that the two-dimensional torus Torus (m, n), m, n ≥ 3, is super spanning connected if at least one of m and n is odd and super spanning laceable if both m and n are even. Furthermore, the super spanning connectivity and spanning laceability of tori with faulty elements have been discussed.

scholarly journals The Partition Dimension of a Path Graph

2021 ◽  
Vol 13 (2) ◽  
pp. 66
Author(s):  
Vivi Ramdhani ◽  
Fathur Rahmi
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Path Graph ◽  
Partition Dimension

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph,  with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

scholarly journals On Average Distance of Neighborhood Graphs and Its Applications

2021 ◽  
Author(s):  
Elias Mwakilama ◽  
Patrick Ali ◽  
Patrick Chidzalo ◽  
Kambombo Mtonga ◽  
Levis Eneya
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Average Distance ◽  
Distance Measures ◽  
Graph Invariants ◽  
Matlab Software ◽  
Software Simulation ◽  
Neighborhood Graph ◽  
Research Questions ◽  
Simulation Results

Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then and μG1′≤μG and μG2′≤μG whenever G1′ and G2′ are components of G′. In order to make our claims substantial and determine graphs for which the bounds are best possible, we performed some experiments in MATLAB software. Simulation results agree very well with the propositions and proofs. Finally, we have described how our results may be applied in socio-epidemiology and ecology and then concluded with other proposed further research questions.

scholarly journals DIMENSI PARTISI GRAF THORN DARI GRAF RODA W3 DAN W4

2019 ◽  
Vol 16 (1) ◽  
pp. 110-115
Author(s):  
R Riza ◽  
S Zayendra ◽  
A Mardhaningsih
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Wheel Graph ◽  
Partition Dimension ◽  
Thorn Graph

Let 𝐺 = (𝑉, 𝐸) be a connected graph and 𝑆 ⊆ 𝑉(𝐺). For a vertex v ∈ V(G) and an ordered k-partition Π = {𝑆1 , 𝑆2 , … , 𝑆𝑘 } of 𝑉(𝐺), the representation of v with respect to Π is the k-vector 𝑟(𝑣|𝛱 = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), . . . , 𝑑(𝑣, 𝑆𝑘)), where d(v,Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if for every two vertices 𝑢, 𝑣  𝑉(𝐺), the representation 𝑟(𝑢|П)  𝑟(𝑣|Π). The minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). The wheel graph 𝑊𝑛 𝑜𝑛 𝑛 + 1 vertices with 𝑉(𝑊𝑛) = {𝑣0, 𝑣1, . . . , 𝑣𝑛}. Let 𝑙2 ,𝑙2 ,… ,𝑙𝑛be non-negative integers, 𝑙𝑖 ≥ 1, for 𝑖  {0,1,2, . . . , 𝑛}. The thorn graph of the graph Wn, with parameters 𝑙0 ,𝑙1 ,… ,𝑙𝑛 is obtained by attaching li new vertices of degree one to the vertex vi of the graph Wn. The thorn graph is denoted by 𝑇ℎ(𝑊𝑛,𝑙0 ,𝑙1 ,… ,𝑙𝑛). In this paper we give the upper bounds for the partition dimension of 𝑊3 and 𝑊4 denoted by 𝑝𝑑(𝑇ℎ(𝑊3 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 )) and 𝑝𝑑(𝑇ℎ(𝑊4 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 ,𝑙4 )). Keywords : Partition Dimension, Resolving Partition, Thorn Graph, Wheel Graph.

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 Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

On fault-tolerant partition dimension of graphs

2021 ◽  
Vol 40 (1) ◽  
pp. 1129-1135
Author(s):  
Kamran Azhar ◽  
Sohail Zafar ◽  
Agha Kashif ◽  
Zohaib Zahid
Keyword(s):  
Connected Graph ◽  
Intelligent Systems ◽  
Fault Tolerant ◽  
Robot Navigation ◽  
Natural Extension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Sensor Networking ◽  
Partition Dimension

Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

scholarly journals Partition Dimension of Dutch Windmill Graph

2021 ◽  
Vol 17 (3) ◽  
pp. 472-483
Author(s):  
Hasmawati Hasmawati ◽  
Budi Nurwahyu ◽  
Ahmad Syukur Daming ◽  
Amir Kamal Amir
Keyword(s):  
Connected Graph ◽  
Partition Dimension

Let be a connected graph G and -partition of  end . The coordinat to  is definition . If every twovertex is distinct  applies, then  is a called  partition of . The minimum k for which k-resolving partition of  is the partition dimension  and denoted with . In this paper, we investigates the partition dimensionfor a large Dutch windmill graph  for  and . We show that if  for some, forany.

scholarly journals Wiener-type invariants and Hamiltonian properties of graphs

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4045-4058
Author(s):  
Qiannan Zhou ◽  
Ligong Wang ◽  
Yong Lu
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
General Graph ◽  
Wiener Type ◽  
Hamilton Connected ◽  
Hamiltonian Properties ◽  
Simple Connected Graph

The Wiener-type invariants of a simple connected graph G = (V(G), E(G)) can be expressed in terms of the quantities Wf = ? {u,v}?V(G)f(dG(u,v)) for various choices of the function f(x), where dG(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a bipartite graph to be Hamiltonian or a connected general graph to be Hamilton-connected and traceable from every vertex in terms of the Wiener-type invariants of G or the complement of G.

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