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 Graceful Labeling for Disconnected Grid Related Graphs

2015 ◽  
Vol 11 ◽  
pp. 6-11 ◽  
Author(s):  
V.J. Kaneria ◽  
H.M. Makadia ◽  
R.V. Viradia
Keyword(s):  
Grid Graph ◽  
Graceful Labeling ◽  
Grid Graphs

In this paper we have proved that union of three grid graphs, U3l=1(Pnl×Pml)and union of finite copies of a grid graph (Pn×Pm)are graceful. We have also given two graceful labeling functions to the grid graph (Pn×Pm).

scholarly journals Edge Even Graceful Labeling of Polar Grid Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 38 ◽  
Author(s):  
Salama Nagy Daoud
Keyword(s):  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Grid Graph ◽  
Graceful Labeling ◽  
Grid Graphs ◽  
Polar Grid ◽  
Necessary And Sufficient

Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , 4 , … , 2 q } such that, when each vertex is assigned the sum of all edges incident to it mod 2 r where r = max { p , q } , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph.

Visual/Graphic Aids for the Technical Report

1979 ◽  
Vol 9 (3) ◽  
pp. 287-291 ◽  
Author(s):  
Robert Cury
Keyword(s):  
Flow Chart ◽  
White Space ◽  
Grid Graph ◽  
Grid Graphs ◽  
Process Maps ◽  
Technical Report ◽  
Technical Writer ◽  
Flow Charts ◽  
Visual Description ◽  
Visual Graphic

Authors of technical papers have many visual/graphic aids available to them. The most common are: grid graphs, tables, bar charts, flow charts, maps, pie diagrams, and drawings and sketches. Grid graphs are used to show relationships. Tables allow the reader to make comparisons of data. The bar chart is another form of the grid graph and is used for the same purpose. A flow chart gives the reader a visual description of a process. Maps show the location of specific features. Pie diagrams show the proportional breakdown of a topic. Pictures and sketches show the reader exactly what is being talked about in the report. Visual/graphic aids allow the technical writer to condense and present his information in an aesthetically pleasing manner; in addition, these aids serve as psychological white space.

scholarly journals ON MAXIMAL ENERGY AND HOSOYA INDEX OF TREES WITHOUT PERFECT MATCHING

2009 ◽  
Vol 81 (1) ◽  
pp. 47-57 ◽  
Author(s):  
HONGBO HUA
Keyword(s):  
Maximal Element ◽  
Perfect Matching ◽  
Undirected Graph ◽  
Unique Element ◽  
Hosoya Index ◽  
Maximal Energy ◽  
Molecular Graphs ◽  
The Family ◽  
Appl Math ◽  
Chemical Trees

AbstractLet G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

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 Computing the Domination Number of Grid Graphs

10.37236/628 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Samu Alanko ◽  
Simon Crevals ◽  
Anton Isopoussu ◽  
Patric Östergård ◽  
Ville Pettersson
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Square Grid ◽  
Minimum Dominating Set

Let $\gamma_{m,n}$ denote the size of a minimum dominating set in the $m \times n$ grid graph. For the square grid graph, exact values for $\gamma_{n,n}$ have earlier been published for $n \leq 19$. By using a dynamic programming algorithm, the values of $\gamma_{m,n}$ for $m,n \leq 29$ are here obtained. Minimum dominating sets for square grid graphs up to size $29 \times 29$ are depicted.

scholarly journals Graceful Labeling of Hypertrees

2021 ◽  
Vol 13 (1) ◽  
pp. 28
Author(s):  
H. El-Zohny ◽  
S. Radwan ◽  
S.I. Abo El-Fotooh ◽  
Z. Mohammed
Keyword(s):  
Graph Theory ◽  
Simple Graph ◽  
Graph Labeling ◽  
Graceful Labeling ◽  
Edge Labeling

Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G  (vertex labeling) or to edges of G  (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices’ labels of each edge is distinct and each one is the label of the corresponding edge.

scholarly journals Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Mrinal Nandi ◽  
Subrata Parui ◽  
Avishek Adhikari
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Domination Number ◽  
Cartesian Product ◽  
Wireless Sensor ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Cylindrical Grid ◽  
Domination Numbers

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

scholarly journals Combinatorics of one-dimensional simple Toeplitz subshifts

2018 ◽  
Vol 40 (6) ◽  
pp. 1673-1714
Author(s):  
DANIEL SELL
Keyword(s):  
Systematic Study ◽  
Henri Poincaré ◽  
Explicit Formulas ◽  
De Bruijn Graphs ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
Complexity Function ◽  
Schreier Graphs ◽  
The Family ◽  
De Bruijn

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

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.

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