scholarly journals On Partition Dimension of Some Cycle-Related Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Changcheng Wei ◽  
Muhammad Faisal Nadeem ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Muhammad Azeem ◽  
Jia-Bao Liu ◽  
...  
Keyword(s):  
Connected Graph ◽  
Partition Dimension ◽  
Simple Connected Graph ◽  
Cycle Graph

Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α  w . r . t  Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

scholarly journals On bounded partition dimension of different families of convex polytopes with pendant edges

2021 ◽  
Vol 7 (3) ◽  
pp. 4405-4415
Author(s):  
Adnan Khali ◽  
◽  
Sh. K Said Husain ◽  
Muhammad Faisal Nadeem ◽  
◽  
...  
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Convex Polytopes ◽  
General Family ◽  
Minimum Value ◽  
Partition Dimension ◽  
Simple Connected Graph

<abstract><p>Let $ \psi = (V, E) $ be a simple connected graph. The distance between $ \rho_1, \rho_2\in V(\psi) $ is the length of a shortest path between $ \rho_1 $ and $ \rho_2. $ Let $ \Gamma = \{\Gamma_1, \Gamma_2, \dots, \Gamma_j\} $ be an ordered partition of the vertices of $ \psi $. Let $ \rho_1\in V(\psi) $, and $ r(\rho_1|\Gamma) = \{d(\rho_1, \Gamma_1), d(\rho_1, \Gamma_2), \dots, d(\rho_1, \Gamma_j)\} $ be a $ j $-tuple. If the representation $ r(\rho_1|\Gamma) $ of every $ \rho_1\in V(\psi) $ w.r.t. $ \Gamma $ is unique then $ \Gamma $ is the resolving partition set of vertices of $ \psi $. The minimum value of $ j $ in the resolving partition set is known as partition dimension and written as $ pd(\psi). $ The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as $ R_n^P $, $ D_n^p $ and $ Q_n^p $ and proved that these graphs have bounded partition dimension.</p></abstract>

scholarly journals Orthogonal labeling

2016 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Bernard Immanuel ◽  
Kiki A. Sugeng
Keyword(s):  
Euclidean Space ◽  
Early Result ◽  
Algebraic Structure ◽  
Connected Graph ◽  
Maximum Degree ◽  
Injective Mapping ◽  
Distance Magic Labeling ◽  
Harmonious Labeling ◽  
Simple Connected Graph ◽  
Cycle Graph

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let ∆</span><span>G </span><span>be the maximum degree of a simple connected graph </span><span>G</span><span>(</span><span>V,E</span><span>). An injective mapping </span><span>P </span><span>: </span><span>V </span><span>→ </span><span>R</span><span>∆</span><span>G </span><span>is said to be an orthogonal labeling of </span><span>G </span><span>if </span><span>uv,uw </span><span>∈ </span><span>E </span><span>implying (</span><span>P</span><span>(</span><span>v</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) </span><span>· </span><span>(</span><span>P</span><span>(</span><span>w</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) = 0, where </span><span>· </span><span>is the usual dot product defined in Euclidean space. A graph </span><span>G </span><span>which has an orthogonal labeling is called an orthogonal graph. This labeling is motivated by the existence of several labelings defined by some algebraic structure, i.e. harmonious labeling and group distance magic labeling. In this paper we study some preliminary results on orthogonal labeling. One of the early result is the fact that cycle graph with even vertices are orthogonal, while ones with odd vertices are not. The main results in this paper state that any graph containing </span><span>K</span><span>3 </span><span>as its subgraph is non-orthogonal and that a graph </span><span>G</span><span>′ </span><span>obtained from adding a pendant to a vertex in orthogonal graph </span><span>G </span><span>is orthogonal. In the end of the paper we state the corollary that any tree is orthogonal.<br /> </span></p></div></div></div>

scholarly journals Dimensi Partisi Graf Hasil Amalgamasi Siklus

2019 ◽  
Vol 16 (2) ◽  
pp. 199
Author(s):  
Hasmawati Hasmawati ◽  
Ahmad Syukur Daming ◽  
Loeky Haryanto ◽  
Budi Nurwahyu
Keyword(s):  
Connected Graph ◽  
Mathematical Induction ◽  
Induction Method ◽  
Partition Dimension ◽  
Cycle Graph

Let be a connected graph G and -partition  of  end . The coordinat  to  is definition . If   for every two vertices in t , then  is a called  k-resolving partition of . The minimum k such that  is a k-resolving partition of  is the partition dimension of  and denoted by . In this paper, we show that the partition dimension for amlagamation of cycle graph  for To proof this results, we was used mathematical induction method. 

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

2007 ◽  
Vol 3 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

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

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

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