scholarly journals On SD-Prime Labelings of Some One Point Unions of Gears

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 849
Author(s):  
Wai-Chee Shiu ◽  
Gee-Choon Lau
Keyword(s):  
Undirected Graph ◽  
Prime Labeling ◽  
Edge Labeling

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f:V(G)→{1,…,n}, and every edge uv in E(G), let S=f(u)+f(v) and D=|f(u)−f(v)|. The labeling f induces an edge labeling f′:E(G)→{0,1} such that for an edge uv in E(G), f′(uv)=1 if gcd(S,D)=1, and f′(uv)=0 otherwise. Such a labeling is called an SD-prime labeling if f′(uv)=1 for all uv∈E(G). We provide SD-prime labelings for some one point unions of gear graphs.

scholarly journals Power-3 Heronian Mean Labeling of Graphs

2020 ◽  
Vol 9 (4) ◽  
pp. 1359-1361
Keyword(s):  
Undirected Graph ◽  
Heronian Mean ◽  
Edge Labeling

Let be an undirected graph having vertices and edges. Now, defining a function say, is called Power-3 Heronian Mean Labeling of a graph if we could able to label the vertices with dissimilar elements from such that it induces an edge labeling defined as, is dissimilar for all the edges (i,e.) It intimates that the dissimilar vertex labeling induces a dissimilar edge labeling on the graph. The graph which owns Power-3 Heronian Mean Labeling is called an Power-3 Heronian Mean Graph. In this, we have advocated the Power-3 Heronian Mean Labeling of some standard graphs like Path, Comb, Caterpillar, Triangular Snake, Quadrilateral Snake and Ladder.

scholarly journals Edge Even Graceful Labeling of Cylinder Grid Graph

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 584 ◽  
Author(s):  
Ahmed A. Elsonbaty ◽  
Salama Nagy Daoud
Keyword(s):  
Undirected Graph ◽  
Grid Graph ◽  
Explicit Formulas ◽  
Graceful Labeling ◽  
Grid Graphs ◽  
The Family ◽  
Edge Labeling

Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order P = | ( V ( G ) | and size q = | E ( G ) | is a bijection f : E ( G ) → { 2 , 4 , … , 2 q } , such that when each vertex v ∈ V ( G ) is assigned the modular sum of the labels (images of f ) of the edges incident to v , the resulting vertex labels are distinct mod 2 r , where r = max ( p , q ) . In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven.

scholarly journals On a Question of Prime Labeling of Graphs

2021 ◽  
pp. 87-93
Author(s):  
A. M. C. U. M. Athapattu ◽  
P. G. R. S. Ranasinghe
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Undirected Graph ◽  
Prime Labeling

In the field of graph theory, the complete graph  of  vertices is a simple undirected graph such that every pair of distinct vertices is connected by a unique edge. In the present work, we introduce planar subgraph  of  with maximal number of edges . Generally,  does not admit prime labeling. We present an algorithm to obtain prime-labeled subgraphs of  . We conclude the paper by stating two conjectures based on labeling of . In particular, the planar subgraph admits anti-magic labeling but does not admit edge magic total labeling.

scholarly journals On b-edge consecutive edge labeling of some regular tree

2020 ◽  
Vol 4 (1) ◽  
pp. 76
Author(s):  
Kiki Ariyanti Sugeng ◽  
Denny R. Silaban
Keyword(s):  
Undirected Graph ◽  
Title Page ◽  
Regular Tree ◽  
Regular Path ◽  
Edge Labeling

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em> E</em></span><span>) </span><span>be a finite (non-empty), simple, connected and undirected graph, where </span><span>V </span><span>and </span><span>E </span><span>are the sets of vertices and edges of </span><span>G</span><span>. An edge magic total labeling is a bijection </span><span>α </span><span>from </span><span><em>V</em> </span><span>∪ </span><span><em>E</em> </span><span>to the integers </span><span>1</span><span>, </span><span>2</span><span>, . . . , <em>n</em> </span><span>+ </span><em>e</em><span>, with the property that for every </span><span><em>xy</em> </span><span>∈ </span><em>E</em><span>, </span><span>α</span><span>(</span><em>x</em><span>) + </span><span>α</span><span>(</span><em>y</em><span>) + </span><span>α</span><span>(</span><em>xy</em><span>) = </span><em>k</em><span>, for some constant </span><em>k</em><span>. Such a labeling is called a </span><em>b</em><span>-edge consecutive edge magic total if </span><span>α</span><span>(</span><em>E</em><span>) = </span><span>{</span><span><em>b</em> </span><span>+ 1</span><span>, <em>b</em> </span><span>+ 2</span><span>, . . . , <em>b</em> </span><span>+ </span><em>e</em><span>}</span><span>. In this paper, we proved that several classes of regular trees, such as regular caterpillars, regular firecrackers, regular caterpillar-like trees, regular path-like trees, and regular banana trees, have a </span><em>b</em><span>-edge consecutive edge magic labeling for some </span><span>0 </span><span>&lt; <em>b</em> &lt; </span><span>|</span><span><em>V</em> </span><span>|</span><span>.</span></p></div></div></div>

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

scholarly journals Common closed neighbourhood prime labeling

2021 ◽  
Vol 1836 (1) ◽  
pp. 012009
Author(s):  
Rinurwati ◽  
A S Alfiyani
Keyword(s):  
Prime Labeling

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

Two Streamlined Depth-First Search Algorithms

1986 ◽  
Vol 9 (1) ◽  
pp. 85-94
Author(s):  
Robert Endre Tarjan
Keyword(s):  
Directed Graph ◽  
Graph Algorithms ◽  
Undirected Graph ◽  
Linear Time ◽  
Search Algorithms ◽  
Depth First Search ◽  
Time Graph

Many linear-time graph algorithms using depth-first search have been invented. We propose simplified versions of two such algorithms, for computing a bipolar orientation or st-numbering of an undirected graph and for finding all feedback vertices of a directed graph.

MINIMUM CONNECTED r-HOP k-DOMINATING SET IN WIRELESS NETWORKS

2009 ◽  
Vol 01 (01) ◽  
pp. 45-57 ◽  
Author(s):  
DEYING LI ◽  
LIN LIU ◽  
HUIQIANG YANG
Keyword(s):  
Wireless Networks ◽  
Undirected Graph ◽  
Dominating Set ◽  
Approximation Ratio ◽  
Dominating Set Problem ◽  
Simulation Results

In this paper, we study the connected r-hop k-dominating set problem in wireless networks. We propose two algorithms for the problem. We prove that algorithm I for UDG has (2r + 1)3 approximate ratio for k ≤ (2r + 1)2 and (2r + 1)((2r + 1)2 + 1)-approximate ratio for k > (2r + 1)2. And algorithm II for any undirected graph has (2r + 1) ln (Δr) approximation ratio, where Δr is the largest cardinality among all r-hop neighborhoods in the network. The simulation results show that our algorithms are efficient.

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