scholarly journals Super (a, d)-Edge Antimagic Total Labeling of Connected Ferris Wheel Graph

2015 ◽  
Vol 15 (2) ◽  
pp. 123
Author(s):  
Djoni Budi Sumarno ◽  
D Dafik ◽  
Kiswara Agung Santoso
Keyword(s):  
Key Words ◽  
Simple Graph ◽  
Axiomatic Method ◽  
Arithmetic Sequence ◽  
Wheel Graph ◽  
Wheel Graphs ◽  
Ferris Wheel

Let G be a simple graph of order p and size q. Graph G is called an (a,d)-edge-antimagic totalifthereexistabijectionf :V(G)∪E(G)→{1,2,...,p+q}suchthattheedge-weights,w(uv)= f(u)+f(v)+f(uv); u, v ∈ V (G), uv ∈ E(G), form an arithmetic sequence with first term a and common difference d. Such a graph G is called super if the smallest possible labels appear on the vertices. In this paper we study super (a, d)-edge antimagic total properties of connected of Ferris Wheel F Wm,n by using deductive axiomatic method. The results of this research are a lemma or theorem. The new theorems show that a connected ferris wheel graphs admit a super (a, d)-edge antimagic total labeling for d = 0, 1, 2. It can be concluded that the result of this research has covered all feasible d. Key Words : (a, d)-edge antimagic vertex labeling, super (a, d)-edge antimagic total labeling, Ferris Wheel graph FWm,n.  

The comparative analysis of metric and edge metric dimension of some subdivisions of the wheel graph

2020 ◽  
pp. 2150062
Author(s):  
Zahid Raza ◽  
M. S. Bataineh
Keyword(s):  
Comparative Analysis ◽  
Metric Dimension ◽  
Wheel Graph ◽  
Wheel Graphs ◽  
Metric Dimensions ◽  
Appl Math

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which [Formula: see text], and also [Formula: see text] which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math. 251 (2018) 204–220].

scholarly journals On the Connected Safe Number of Some Classes of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Rakib Iqbal ◽  
Muhammad Shoaib Sardar ◽  
Dalal Alrowaili ◽  
Sohail Zafar ◽  
Imran Siddique
Keyword(s):  
Connected Graph ◽  
Nonempty Subset ◽  
Simple Graph ◽  
Minimal Cardinality ◽  
Connected Component ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Wheel Graphs ◽  
Safe Set

For a connected simple graph G , a nonempty subset S of V G is a connected safe set if the induced subgraph G S is connected and the inequality S ≥ D satisfies for each connected component D of G∖S whenever an edge of G exists between S and D . A connected safe set of a connected graph G with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.

Connectivity in Matroids

1966 ◽  
Vol 18 ◽  
pp. 1301-1324 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Single Vertex ◽  
Wheel Graph ◽  
Wheel Graphs

An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only 3-connected graphs in which every edge is essential are the “wheel-graphs.” A wheel-graph of order n, where n is an integer ⩾3, is constructed from an n-gon called its “rim” by adding one new vertex, called the “hub,” and n new edges, or “spokes” joining the new vertex to the n vertices of the rim; see Figure 4A.A matroid can be regarded as a generalized graph. One way of developing the theory of matroids is therefore to generalize known theorems about graphs. In the present paper we do this with the theorem stated above.

scholarly journals On Divisor 3-Equitable Labeling of Wheel Graphs

2020 ◽  
Vol 8 (5) ◽  
pp. 5480-5483
Keyword(s):  
Complete Graph ◽  
Single Vertex ◽  
Wheel Graph ◽  
Wheel Graphs

A graph 𝑮 on 𝒏 vertices is said to admit a divisor 3- equitable labeling if there exists a bijection 𝒅 ∶ 𝑽(𝑮) → {𝟏, 𝟐, . . . , 𝒏} defined by 𝒅 𝒆 = 𝒙𝒚 = 𝟏,𝒊𝒇𝒅(𝒙)|𝒅 𝒚 or 𝒅 𝒚 |𝒅(𝒙) 𝟐, 𝒊𝒇 𝒅 𝒙 𝒅 𝒚 = 𝟐 𝒐𝒓 𝒅 𝒚 𝒅 𝒙 = 𝟐 𝟎, 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆 and |𝒆𝒅 𝒊 − 𝒆𝒅 𝒋 | ≤ 𝟏 for all 𝟎 ≤ 𝒊,𝒋 ≤ 𝟐, where 𝒆𝒅 𝒊 denotes the number of edges labelled with “𝒊”. A graph which permits a divisor 3-equitable labeling is called a divisor 3-equitable graph. A wheel graph 𝑾𝒏 is defined as 𝑾𝒏 = 𝑪𝒏−𝟏⋀ 𝑲𝟏, where 𝑪𝒏−𝟏 is a cycle on 𝒏 − 𝟏 vertices and 𝑲𝟏 is a complete graph on a single vertex. In this paper, we prove the non-existence of a divisor 3- equitable labeling of the wheel graph 𝑾𝒏 for 𝒏 ≥ 𝟕.

scholarly journals Linear Time Recognition of P4-Indifferent Graphs

1999 ◽  
Vol 6 (38) ◽  
Author(s):  
Romeo Rizzi
Keyword(s):  
Key Words ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Modular Decomposition ◽  
Forbidden Induced Subgraphs ◽  
Chordless Path

<p>A simple graph is P4-indifferent if it admits a total order < on<br />its nodes such that every chordless path with nodes a, b, c, d and edges<br />ab, bc, cd has a < b < c < d or a > b > c > d. P4-indifferent graphs generalize<br /> indifferent graphs and are perfectly orderable. Recently, Hoang,<br />Maray and Noy gave a characterization of P4-indifferent graphs in<br />terms of forbidden induced subgraphs. We clarify their proof and describe<br /> a linear time algorithm to recognize P4-indifferent graphs. When<br />the input is a P4-indifferent graph, then the algorithm computes an order < as above.</p><p>Key words: P4-indifference, linear time, recognition, modular decomposition.</p><p> </p>

scholarly journals The Vertex-Edge Resolvability of Some Wheel-Related Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Bao-Hua Xing ◽  
Sunny Kumar Sharma ◽  
Vijay Kumar Bhat ◽  
Hassan Raza ◽  
Jia-Bao Liu
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Wheel Graph ◽  
Wheel Graphs ◽  
Metric Dimensions ◽  
Mixed Metric

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m  dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

scholarly journals Total Face Irregularity Strength of Grid and Wheel Graph under K-Labeling of Type (1, 1, 0)

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Aleem Mughal ◽  
Noshad Jamil
Keyword(s):  
Plane Graph ◽  
Plane Graphs ◽  
Wheel Graph ◽  
Wheel Graphs ◽  
The Face ◽  
Vertex Set ◽  
Positive Integers ◽  
Special Case ◽  
Irregularity Strength

In this study, we used grids and wheel graphs G = V , E , F , which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k -labeling of type α , β , γ . In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer k for which a total label of all verteices and edges of a plane graph has distinct face weights is called k -labeling of a graph. The integer k is named as total face irregularity strength of the graph and denoted as tfs G . We also discussed a special case of total face irregularity strength of plane graphs under k -labeling of type (1, 1, 0). The results will be verified by using figures and examples.

scholarly journals Super (a, d)-H-antimagic labeling of subdivided graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 688-697
Author(s):  
Amir Taimur ◽  
Muhammad Numan ◽  
Gohar Ali ◽  
Adeela Mumtaz ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Arithmetic Sequence ◽  
Antimagic Labeling ◽  
Bijective Function

AbstractA simple graphG= (V,E) admits anH-covering, if every edge inE(G) belongs to a subgraph ofGisomorphic toH. A graphGadmitting anH-covering is called an (a,d)-H-antimagic if there exists a bijective functionf:V(G) ∪E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphsH′ isomorphic toHthe sums ∑v∈V(H′)f(v) + ∑e∈E(H′)f(e) form an arithmetic sequence {a,a+d, …,a+ (t− 1)d}, wherea> 0 andd≥ 0 are integers andtis the number of all subgraphs ofGisomorphic toH. Moreover, if the vertices are labeled with numbers 1, 2, …, |V(G)| the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a,d)-cycle-antimagic labeling for somed.

Ferris Wheel Graphs

2019 ◽  
Vol 112 (7) ◽  
pp. 560
Author(s):  
Wayne Nirode
Keyword(s):  
Wheel Graphs ◽  
Sinusoidal Functions ◽  
Ferris Wheel

To introduce sinusoidal functions, I use an animation of a Ferris wheel rotating for 60 seconds, with one seat labeled You (see fig. 1). Students draw a graph of their height above ground as a function of time with appropriate units and scales on both axes. Next a volunteer shares his or her graph. I then ask someone to share a different graph. I choose one student with a curved graph (see fig. 2a) and another with a piece-wise linear (sawtooth) graph (see fig. 2b).

scholarly journals Congruence Classes of Orientable $2$-Cell Embeddings of Bouquets of Circles and Dipoles

10.37236/313 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin-Ho Kwak ◽  
Jin-Xin Zhou
Keyword(s):  
Orientable Surface ◽  
Simple Graph ◽  
Complete Graphs ◽  
Graph Automorphism ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs ◽  
Multiple Edges ◽  
Orientable Surfaces ◽  
Surface Homeomorphism ◽  
Congruence Classes

Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.

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