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 Schultz and Modified Schultz Polynomials of Coronene Polycyclic Aromatic Hydrocarbons

2014 ◽  
Vol 32 ◽  
pp. 1-10
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Polycyclic Aromatic Hydrocarbons ◽  
Graph Theory ◽  
Shortest Path ◽  
Aromatic Hydrocarbons ◽  
Connected Graph ◽  
Molecular Graph ◽  
Hydrocarbon Molecule ◽  
Polycyclic Aromatic ◽  
Hosoya Polynomial ◽  
Simple Connected Graph

Let G = (V;E) be a simple connected graph. The sets of vertices and edges of G are denoted byV = V(G) and E = E(G), respectively. In such a simple molecular graph, vertices represent atoms andedges represent bonds. The distance between the vertices u and v in V(G) of graph G is the number ofedges in a shortest path connecting them, we denote by d(u,v). In graph theory, we have manyinvariant polynomials for a graph G. In this research, we computing the Schultz polynomial, ModifiedSchultz polynomial, Hosoya polynomial and their topological indices of a Hydrocarbon molecule, thatwe call “Coronene Polycyclic Aromatic Hydrocarbons”.

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.

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 GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

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

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.

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