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Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs
CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 4 (4) (2017), Pages 161-166 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs Fery Firmansah1, Muhammad Ridlo Yuwono2 1, 2Mathematics Edu. Depart. University of Widya Dharma Klaten, Indonesia Email: [email protected], [email protected] ABSTRACT A graph 퐺(푉(퐺), 퐸(퐺)) is called graph 퐺(푝, 푞) if it has 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. The graph 퐺(푝, 푞) is said to be odd harmonious if there exist an injection 푓: 푉(퐺) → {0,1,2, … ,2푞 − 1} such that the induced function 푓∗: 퐸(퐺) → {1,3,5, … ,2푞 − 1} defined by 푓∗(푢푣) = 푓(푢) + 푓(푣). The function 푓∗ is a bijection and 푓 is said to be odd harmonious labeling of 퐺(푝, 푞). In this paper we prove that pleated of the (푘) Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 are odd harmonious graph. Moreover, we also give odd (푘) (푘) harmonious labeling construction for the union pleated of the Dutch windmill graph 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1. Keywords: odd harmonious labeling, pleated graph, the Dutch windmill graph INTRODUCTION In this paper we consider simple, finite, connected and undirected graph. A graph 퐺(푝, 푞) with 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. A graph labeling which has often been motivated by practical problems is one of fascinating areas of research. Labeled graphs serves as useful mathematical models for many applications in coding theory, communication networks, and mobile telecommunication system. -
Strong Edge Graceful Labeling of Windmill Graphs ∑
International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 1 (2013), pp. 19-26 © International Research Publication House http://www.irphouse.com Strong Edge Graceful Labeling of Windmill Graphs Dr. M. Subbiah VKS College Of Engineering& Technology Desiyamangalam, Karur - 639120 [email protected]. Abstract A (p, q) graph G is said to have strong edge graceful labeling if there 3q exists an injection f from the edge set to 1,2, ... so that the 2 induced mapping f+ defined on the vertex set given by f x fxy xy EG mod 2p are distinct. A graph G is said to be strong edge graceful if it admits a strong edge graceful labeling. In this paper we investigate strong edge graceful labeling of Windmill graph. (n) Definition: The windmill graphs Km (n >3) to be the family of graphs consisting of n copies of Km with a vertex in common. (n) Theorem: 1. The windmill graph K4 is strong edge graceful for all n 3 when n is even. (n) Proof: Let {v1, v2, v3, ..., v3n, } be the vertices of K4 and {e1, e2, e3, ...,e3n- (n) 1, e3n, , f1, ,f2, ,f3, . .f3n-1, f3n. } be the edges of K4 which are denoted as in the following Fig. 1. 20 Dr. M. Subbiah . v e 3 3 n n -1 . -1 . v . 3 n f . 3 n -2 f 3 e n . 3 -1 n e 3 2 n n -2 f v v 3 3n 0 2 v 3 . n-2 f v 1 . 2 f 2 f 3 f . -
Ssp Structure of Some Graph Classes
International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 939-948 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: A http://www.ijpam.eu P ijpam.eu SSP STRUCTURE OF SOME GRAPH CLASSES R. Mary Jeya Jothi1, A. Amutha2 1,2Department of Mathematics Sathyabama University Chennai, 119, INDIA Abstract: A Graph G is Super Strongly Perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. Strongly perfect, complete bipartite graph, etc., are some of the most important classes of super strongly perfect graphs. Here, we analyse the other classes of super strongly perfect graphs like trivially perfect graphs and very strongly perfect graphs. We investigate the structure of super strongly perfect graph in friendship graphs. Also, we discuss the maximal cliques, colourability and dominating sets in friendship graphs. AMS Subject Classification: 05C75 Key Words: super strongly perfect graph, trivially perfect graph, very strongly perfect graph and friendship graph 1. Introduction Super strongly perfect graph is defined by B.D. Acharya and its characterization has been given as an open problem in 2006 [5]. Many analysation on super strongly perfect graph have been done by us in our previous research work, (i.e.,) we have investigated the classes of super strongly perfect graphs like strongly c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 940 R.M.J. Jothi, A. Amutha perfect, complete bipartite graph, etc., [3, 4]. In this paper we have discussed some other classes of super strongly perfect graphs like trivially perfect graph, very strongly perfect graph and friendship graph. -
5Th Lecture : Modular Decomposition MPRI 2013–2014 Schedule A
5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Schedule 5th Lecture : Modular decomposition Introduction MPRI 2013–2014 Graph searches Michel Habib Applications of LBFS on structured graph classes [email protected] http://www.liafa.univ-Paris-Diderot.fr/~habib Chordal graphs Cograph recognition Sophie Germain, 22 octobre 2013 A nice conjecture 5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Introduction A hierarchy of graph models 1. Undirected graphs (graphes non orient´es) 2. Tournaments (Tournois), sometimes 2-circuits are allowed. 3. Signed graphs (Graphes sign´es) each edge is labelled + or - (for example friend or enemy) Examen le mardi 26 novembre de 9h `a12h 4. Oriented graphs (Graphes orient´es), each edge is given a Salle habituelle unique direction (no 2-circuits) An interesting subclass are the DAG Directed Acyclic Graphs (graphes sans circuit), for which the transitive closure is a partial order (ordre partiel) 5. Partial orders and comparability graphs an intersting particular case. Duality comparability – cocomparability (graphes de comparabilit´e– graphes d’incomparabilit´e) 6. Directed graphs or digraphs (Graphes dirig´es) 5th Lecture : Modular decomposition MPRI 2013–2014 5th Lecture : Modular decomposition MPRI 2013–2014 Introduction Introduction The problem has to be defined in each model and sometimes it could be hard. ◮ What is the right notion for a coloration in a directed graph ? For partial orders, comparability graphs or uncomparability graphs ◮ No directed cycle unicolored, seems to be the good one. the independant set and maximum clique problems are polynomial. ◮ It took 20 years to find the right notion of oriented matro¨ıd ◮ What is the right notion of treewidth for directed graphs ? ◮ Still an open question. -
OPTIMAL SOLUTIONS and RANKS in the MAX-CUT SDP Daniel
COVER PAGE: OPTIMAL SOLUTIONS AND RANKS IN THE MAX-CUT SDP Daniel Hong, Skyline High School, Sammamish, WA, USA Hyunwoo Lee, Interlake High School, Bellevue, WA, USA Alex Wei, Interlake High School, Bellevue, WA, USA Instructor: Diego Cifuentes, Massachusetts Institute of Technology, Cam- bridge, MA, USA 1 OPTIMAL SOLUTIONS AND RANKS IN THE MAX-CUT SDP DANIEL HONG, HYUNWOO LEE, AND ALEX WEI Abstract. The max-cut problem is a classical graph theory problem which is NP-complete. The best polynomial time approximation scheme relies on semidefinite programming (SDP). We study the conditions under which graphs of certain classes have rank 1 solutions to the max-cut SDP. We apply these findings to look at how solutions to the max-cut SDP behave under simple combinatorial constructions. Our results determine when solutions to the max-cut SDP for cycle graphs is rank 1. We find the solutions to the max-cut SDP of the vertex sum of two graphs. We then characterize the SDP solutions upon joining two triangle graphs by an edge sum. Keywords: Max-cut, Semidefinite program, Rank, Cycle graphs, Ver- tex sum, Edge sum Contents 1. Introduction 2 Related work 3 Structure 3 2. Background 4 2.1. Semidefinite programs 4 2.2. Max-cut SDP 4 2.3. Clique sums of graphs 6 3. Experiments on rank of max-cut SDP 6 4. Solutions and Ranks for Particular Classes of Graphs 7 4.1. Rank 1 solutions for cycles 7 4.2. Max-cut SDP for a vertex sum 8 4.3. Max-cut SDP for a diamond graph 9 5. -
Graphic Sequences with a Realization Containing a Generalized Friendship Graph ∗
Graphic Sequences with a Realization Containing a Generalized Friendship Graph ∗ Jian-Hua Yina,† Gang Chenb, John R. Schmittc aDepartment of Applied Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, P.R. China bDepartment of Mathematics, Ningxia University, Yinchuan 750021, P.R. China cDepartment of Mathematics, Middlebury College, Middlebury, VT, USA Abstract: Gould, Jacobson and Lehel (Combinatorics, Graph Theory and Algorithms, Vol.I (1999) 451–460) considered a variation of the classical Tur´an-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H, n) such that every n-term graphic sequence π = (d1, d2, . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥ σ(H, n) has a realization G containing H as a subgraph. Let Ft,r,k denote the generalized friendship graph on kt − kr + r vertices, that is, the graph of k copies of Kt meeting in a common r set, where Kt is the complete graph on t vertices and 0 ≤ r ≤ t. In this paper, we determine σ(Ft,r,k, n) for k ≥ 2, t ≥ 3, 1 ≤ r ≤ t − 2 and n sufficiently large. Keywords: degree sequence, potentially Ft,r,k-graphic sequence, generalized friendship graph. 2000 MR Subject Classification: 05C35, 05C07. 1. Introduction The set of all sequences π = (d1, d2, . , dn) of non-negative, non-increasing integers with d1 ≤ n − 1 is denoted by NSn. A sequence π ∈ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π. -
Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others. -
Algorithms for Generating Star and Path of Graphs Using BFS Dr
Web Site: www.ijettcs.org Email: [email protected], [email protected] Volume 1, Issue 3, September – October 2012 ISSN 2278-6856 Algorithms for Generating Star and Path of Graphs using BFS Dr. H. B. Walikar2, Ravikumar H. Roogi1, Shreedevi V. Shindhe3, Ishwar. B4 2,4Prof. Dept of Computer Science, Karnatak University, Dharwad, 1,3Research Scholars, Dept of Computer Science, Karnatak University, Dharwad Abstract: In this paper we deal with BFS algorithm by 1.8 Diamond Graph: modifying it with some conditions and proper labeling of The diamond graph is the simple graph on nodes and vertices which results edges illustrated Fig.7. [2] and on applying it to some small basic 1.9 Paw Graph: class of graphs. The BFS algorithm has to modify The paw graph is the -pan graph, which is also accordingly. Some graphs will result in and by direct application of BFS where some need modifications isomorphic to the -tadpole graph. Fig.8 [2] in the algorithm. The BFS algorithm starts with a root vertex 1.10 Gem Graph: called start vertex. The resulted output tree structure will be The gem graph is the fan graph illustrated Fig.9 [2] in the form of Structure or in Structure. 1.11 Dart Graph: Keywords: BFS, Graph, Star, Path. The dart graph is the -vertex graph illustrated Fig.10.[2] 1.12 Tetrahedral Graph: 1. INTRODUCTION The tetrahedral graph is the Platonic graph that is the 1.1 Graph: unique polyhedral graph on four nodes which is also the A graph is a finite collection of objects called vertices complete graph and therefore also the wheel graph . -
Interlace Polynomials of Friendship Graphs
Electronic Journal of Graph Theory and Applications 6 (2) (2018), 269–281 Interlace polynomials of friendship graphs Christina Eubanks-Turnera, Aihua Lib aDepartment of Mathematics, Loyola Marymount University, 90045 bDepartment of Mathematical Sciences, Montclair State University, 07043 [email protected], [email protected] Abstract In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdos,¨ Renyi´ and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph. Keywords: graph polynomial, interlace polynomial, friendship graph, butterfly graph Mathematics Subject Classification : 05C31, 05C50 DOI: 10.5614/ejgta.2018.6.2.7 1. Introduction Sequencing by hybridization is a method of reconstructing a long DNA string from its nu- cleotide sequence. Since gaining a unique reconstruction from the substrings is not always possi- ble, a major question that arises in this study is “For a random string, how many reconstructions are possible?” In [2], Arratia, Bollobas,´ Coppersmith, and Sorkin answer an important question related to DNA sequencing by converting this to a question about Euler circuits in a 2-in, 2-out graph that have been “toggled” (interlaced). The previously mentioned authors introduced the in- terlace polynomial of a graph, a polynomial that represents the information gained from doing the toggling process on the graph, see [2]. Interlace polynomials are similar to other graph polynomi- als, such as, Tutte and Martin polynomials, see [5]. Some researchers have studied different types of graph polynomials, such as genus polynomials, [7]. -
Minor-Obstructions for Apex Sub-Unicyclic Graphs1
Minor-Obstructions for Apex Sub-unicyclic Graphs1 Alexandros Leivaditis2 Alexandros Singh3 Giannos Stamoulis3 Dimitrios M. Thilikos4,2,3,5 Konstantinos Tsatsanis3 Vasiliki Velona6,7,8 Abstract A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph G is k-apex sub-unicyclic if it can become sub-unicyclic by removing k of its vertices. We identify 29 graphs that are the minor-obstructions of the class of 1-apex sub-unicyclic graphs, i.e., the set of all minor minimal graphs that do not belong in this class. For bigger values of k, we give an exact structural characterization of all the cactus graphs that are minor-obstructions of k-apex sub- unicyclic graphs and we enumerate them. This implies that, for every k, the class of k-apex sub-unicyclic graphs has at least 0.34 · k−2.5(6.278)k minor-obstructions. Keywords: Graph Minors, Obstruction set, Sub-unicyclc graphs. 1 Introduction A graph is called unicyclic [17] if it contains exactly one cycle and is called sub-unicyclic if it contains at most one cycle. Notice that sub-unicyclic graphs are exactly the subgraphs of unicyclic graphs. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained by some subgraph of G after a series of contractions. We say that a graph class G is minor-closed if every minor of every graph in G also belongs in G. We also define obs(G), called the minor-obstruction set of G, as the set of minor-minimal graphs not in G. -
4-Prime Cordiality of Some Cycle Related Graphs
Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. An International Journal ISSN: 1932-9466 (AAM) Vol. 12, Issue 1 (June 2017), pp. 230 – 240 4-Prime cordiality of some cycle related graphs 1R. Ponraj, 2Rajpal Singh, & 1S. Sathish Narayanan 1Department of Mathematics Sri Paramakalyani College Alwarkurichi-627412, India [email protected]; [email protected] & 2Research Scholar Department of Mathematics Manonmaniam Sundaranar University Abishekapatti, Tirunelveli - 627012, Tamilnadu, India [email protected] Received: June 18, 201; Accepted: October 20, 2016 Abstract Recently three prime cordial labeling behavior of path, cycle, complete graph, wheel, comb, subdivison of a star, bistar, double comb, corona of tree with a vertex, crown, olive tree and other standard graphs were studied. Also four prime cordial labeling behavior of complete graph, book, flower were studied. In this paper, we investigate the four prime cordial labeling behavior of corona of wheel, gear, double cone, helm, closed helm, butterfly graph, and friendship graph. Keywords: Wheel; Gear; Double Cone; Helm; Butterfly Graph MSC 2010 No.: Primary 05C78, Secondary 05C38 1. Introduction Throughout this paper we have considered only simple and undirected graph. Terms and defini- tions not defined here are used in the sense of Harary (2001) and Gallian (2015). 230 AAM: Intern. J., Vol. 12, Issue 1 (June 2017) 231 Let G = (V;E) be a (p;q) graph. The cardinality of V is called the order of G and the cardinality of E is called the size of G. Rosa (1967) introduced graceful labelling of graphs which was the foundation of the graph labelling. -
On the Chromatic Number of (P {5},Windmill)-Free Graphs
Opuscula Math. 37, no. 4 (2017), 609–615 http://dx.doi.org/10.7494/OpMath.2017.37.4.609 Opuscula Mathematica ON THE CHROMATIC NUMBER OF (P5, windmill)-FREE GRAPHS Ingo Schiermeyer Communicated by Hao Li Abstract. In this paper we study the chromatic number of (P5, windmill)-free graphs. For p integers r, p 2 the windmill graph W = K pKr is the graph obtained by joining a single ≥ r+1 1 ∨ vertex (the center) to the vertices of p disjoint copies of a complete graph Kr. Our main result is that every (P5, windmill)-free graph G admits a polynomial χ-binding function. Moreover, we will present polynomial χ-binding functions for several other subclasses of P5-free graphs. Keywords: vertex colouring, perfect graphs, χ-binding function, forbidden induced subgraph. Mathematics Subject Classification: 05C15, 05C17. 1. INTRODUCTION We consider finite, simple, and undirected graphs, and use standard terminology and notation. Let G be a graph. An induced subgraph of G is a graph H such that V (H) V (G), ⊆ and uv E(H) if and only if uv E(G) for all u, v V (H). Given graphs G and F ∈ ∈ ∈ we say that G contains F if F is isomorphic to an induced subgraph of G. We say that a graph G is F -free, if it does not contain F. For two graphs G, H we denote by G + H the disjoint union and by G H the join of G and H, respectively. ∨ A graph G is called k-colourable, if its vertices can be coloured with k colours so that adjacent vertices obtain distinct colours.