Chapter 1. Introduction

Transcription

1 Chapter 1 Introduction 1

2 Chapter 1. Introduction 2 The later part of 20 th century has witnessed unprecedented growth in graph theory. The research activity related to optimization techniques and computer science provided the reason for the development of the subject. Theory of graphs has a close interaction with many branches of mathematics and social sciences like discrete mathematics, network theory, information system and social network analysis. There are many interesting areas of research in graph theory. The labeling of graph is one of such areas which cuts across wide range of applications, especially in coding theory, design of optimal circuit layout and X-ray crystallography as reported in Bloom and Golomb [13] and Yegnanarayanan and Vaidhyanathan [75]. The concept of graph labeling was originated by Rosa [35] in For an extensive survey and bibliographic references on graph labeling we refer to Gallian [19]. The content of the present thesis contains a report of investigations carried out on some graph labeling problems. The content is divided into seven chapters. The current Chapter - 1 is of introductory in nature which provides a brief outline of the thesis. Chapter - 2 is intended to provide terminology and notations needed for the subsequent chapters in order to maintain compactness of the content. Chapter - 3 is focused on product cordial labeling of graphs. We investigate product cordial labeling of some special graphs like Circular Ladder, Möbius Ladder, Step Ladder, Closed Helm, Flower Graph, Gear Graph and Web Graph. Moreover we investigate product cordial labeling for the larger graphs obtained from path and cycle by means of some graph operations. Chapter - 4 is aimed to introduce an edge analogue of product cordial labeling as a variant of cordial labeling. We investigate following results for this newly defined concept.

3 Chapter 1. Introduction 3 Graphs Edge Product Cordial Not Edge Product Cordial Trees Order greater than 2 C n, Gear Graph G n, P 2 n, DT n, Q n, DQ n, DS(S n ), S n odd n even n K n n < 4 n 4 K m,n m,n 2 W n, D 2 (P n ), D 2 (C n ), M(C n ), T (C n ), S (C n ), P n [P 2 ], C n [P 2 ], DS(G n ), all n DS(B n,n ), D f n Crown C n K1, H n, Armed Crown ACr n, CH n, all n Wb n, Fl n, M(P n ), T (P n ), T n S (P n ) even n odd n Unicyclic graphs Odd order m + n even Tadpole C m C (t) n or m + n odd and m > n t even or t and n both odd m + n odd and m < n t odd and n even P m P n, C m C n all m and n P m P n all m and n C m C n m and n even m or n odd Cn 2 n = 3 n 3 DS(P n ) n > 2 n = 2

4 Chapter 1. Introduction 4 In Chapter - 5, we introduce one more variant of cordial labeling and named it as total edge product cordial labeling. We present following results on this newly defined concept. Every edge product cordial graph of either even order or even size admits total edge product cordial labeling. The graphs with degree sequences 1,1, 2,2,2,2 or 3,2,2,1 are not total edge product cordial graphs. K n for n > 2, K m,n except K 1,1 and K 2,2, W n, gear graph G n, C n except for n 4, C (t) n, fan f n and double fan D f n are total edge product cordial graphs. The graphs C 2 n, D 2 (C n ), M(C n ), S (C n ) and T (C n ) admit total edge product cordial labeling. The graphs P 2 n except for n = 2, D 2 (P n ) except for n = 2, M(P n ), S (P n ) and T (P n ) admit total edge product cordial labeling. In penultimate Chapter - 6 we report some new results on geometric and harmonic mean labelings of graphs. The last Chapter - 7 is targeted to discuss embedding and NP-complete problems for some equitable labelings in general. Throughout the thesis some open problems are posed which will provide further scope of research for any beginner. A list of symbols is given just after the table of contents while a bibliography in MLA (Modern Language Association) format is listed in alphabetical order at the end. The work reported in the thesis is also presented in prestigious academic events and the content of the thesis give rise to eighteen research papers which reflects the quality of the work embodied in the thesis. The reprints of the papers published in scholarly, indexed and peer reviewed journals are provided as an annexure.

5 Chapter 1. Introduction 5 List of Publications Arising From the Thesis 1. Product Cordial Labeling for Some New Graphs, Journal of Mathematics Research, 3(2), (2011), ( ISSN: (Print) ISSN: (Online) 2. Some Product Cordial Graphs, Elixir Discrete Mathematics, 41, (2011), ( ISSN: X 3. Product Cordial Graphs in the Context of Some Graph Operations, International Journal of Mathematics and Scientific Computing, 1(2), (2011), 1-6. ( ISSN: Further Results on Product Cordial Graphs, International Journal of Mathematics and Soft Computing, 2(2), (2012), ( ISSN: (Print) ISSN: (Online) Impact Factor: 5.61(Index Copernicus) 5. Further Results on Product Cordial Labeling, International Journal of Mathematical Combinatorics, 3, (2012), ( ISSN:

6 Chapter 1. Introduction 6 6. Product and Edge Product Cordial Labeling of Degree Splitting Graph of Some Graphs, Advances and Applications in Discrete Mathematics, 15(1), (2015), ( ISSN: Impact Factor : Edge Product Cordial Labeling of Graphs, Journal of Mathematical and Computational Science, 2(5), (2012), ( ISSN: Impact Factor = Some Edge Product Cordial Graphs, International Journal of Mathematics and Soft Computing, 3(3), (2013), ( ISSN: (Print) ISSN: (Online) Impact Factor: 5.61(Index Copernicus) 9. Edge Product Cordial Labeling in the Context of Some Graph Operations, International Journal of Mathematics and Scientific Computing, 3(1), (2013), 4-7. ( ISSN: Some New Families of Edge Product Cordial Graphs, Advanced Modeling and Optimization, 15(1), (2013), ( ISSN:

7 Chapter 1. Introduction On Edge Product Cordial Labeling of Some Product Related Graphs, International Journal of Mathematics And Its Applications, 2(2), (2014), ( ISSN: (Online) Impact Factor: Total Edge Product Cordial Labeling of Graphs, Malaya Journal of Matematik, 3(1), (2013), ( ISSN: (Online) ISSN: (Print) Impact Factor: 5.54(Index Copernicus) 13. On Total Edge Product Cordial Labeling, International Journal of Mathematics and Scientific Computing, 3(2), (2013), ( ISSN: Geometric Mean Labeling in the Context of Duplication of Graph Elements, International Journal of Mathematical Sciences and Engineering Applications, 6(6), (2012), ( ISSN: Impact Factor: Harmonic Mean Labeling in the Context of Duplication of Graph Elements, Elixir Discrete Mathematics, 48, (2012), ( ISSN: X

8 Chapter 1. Introduction Embedding and NP-complete Problems for Some Equitable Labelings, Journal of Applied Computer Science & Mathematics, 18(8), (2014), ( ISSN : (Online) ISSN: (Print) 17. On Embedding and NP-Complete Problems of Equitable Labelings, IOSR Journal of Mathematics, 2(1)(III), (2015), ( ISSN: (Online) ISSN: X(Print) 18. Product Cordial Labeling of Line Graph of Some Graphs, communicated for publication. The reprints of published papers are provided as an annexure.

9 Chapter 1. Introduction 9 Details of the Work Presented in Conferences 1. The paper entitled as Product Cordial Labeling for Some New Graphs was presented in Mathematics Meet held at Gujarat University, Ahmedabad (Gujarat) during 3-5 February, The paper entitled as Some Product Cordial Graphs was presented in The Seventh Annual Conference of Academy of Discrete Mathematics and Applications (ADMA) held at National Institute of Technology, Calicut (Kerala) during 9-11 June, The paper entitled as Product Cordial Graphs in the Context of Some Graph Operations was presented in The Eighth Annual Conference of Academy of Discrete Mathematics and Applications (ADMA) held at V. V. Vanniaperumal College for Women, Virudhunagar (Tamil Nadu) during 8-10 June, The paper entitled as Further Results on Product Cordial Graphs was presented in National Conference on Recent Trends in Mathematical Sciences and Their Applications held at Modi Institute of Technology and Science, Sikar (Rajastan) during 5-6 November, The paper entitled as Harmonic Mean Labeling in the Context of Duplication of Graph Elements was presented in National Conference on Advances in Mathematical Sciences held at Kadi Sarva Vishwavidyalay, Gandhinagar (Gujarat) during 7-8 December, The paper entitled as Geometric Mean Labeling in the Context of Duplication of Graph Elements was presented in State Level Conference of Mathematics Celebrating The National Mathematics Year 2012 held at Saurashtra University, Rajkot (Gujarat) on 22 December, The poster entitled as Embedding and NP-complete Problems for Some Equitable Labelings was presented in The Third International Conference on Discrete

10 Chapter 1. Introduction 10 Mathematics (ICDM) held at Karnatak University, Dharwad (Karnatak) during June, The paper entitled as Edge Product Cordial Labeling of Graphs was presented in The Tenth Annual Conference of Academy of Discrete Mathematics and Applications (ADMA) held at Reva Institute of Technology & Management, Bangalore (Karnatak) during June, The paper entitled as On Edge Product Cordial Labeling of Some Product Related Graphs was presented in National Conference on Recent Trends in Mathematics and its Applications held at C. U. Shah University, Wadhwan City (Gujarat) on 19 December, The paper entitled as On Embedding and NP-Complete Problems of Equitable Labelings was presented in State Level Seminar on Advances in Mathematical Sciences at Gujarat Vidhyapith organized by K. K. Shah Jarodwala Maninagar Science College, Ahmedabad (Gujarat) on 14 February, 2015.