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A Study on Color Energy of Graphs A Study on Color Energy of Graphs A Dissertation Submitted in Partial Fulfillment of the Requirements for the Award of the Degree of Master of Philosophy in Mathematics by Prajakta Bharat Joshi (Reg. No. 1640036) Under the Guidance of Mayamma Joseph Professor Department of Mathematics CHRIST UNIVERSITY BENGALURU, INDIA November 2017 Approval of Dissertation Dissertation entitled A Study on Color Energy of Graphs by Prajakta Bharat Joshi, Reg. No. 1640036 is approved for the award of the degree of Master of Philosophy in Mathematics. Examiner: Supervisor: Chairman: GRC: Date: Place: Bengaluru ii DECLARATION I, Prajakta Bharat Joshi, hereby declare that the dissertation, titled A Study on Color Energy of Graphs is a record of original research work undertaken by me for the award of the degree of Master of Philosophy in Mathematics. I have completed this study under the supervision of Dr Mayamma Joseph, Professor, Department of Mathematics. I also declare that this dissertation has not been submitted for the award of any degree, diploma, associateship, fellowship or other title. I hereby confirm the originality of the work and that there is no plagiarism in any part of the dissertation. Place: Bengaluru Date: Prajakta Bharat Joshi Reg. No. 1640036 Department of Mathematics Christ University, Bengaluru. iii CERTIFICATE This is to certify that the dissertation submitted by Prajakta Bharat Joshi, Reg. No. 1640036, titled A Study on Color Energy of Graphs is a record of research work done by her during the academic year 2016-2017 under my supervision in partial fulfillment for the award of Master of Philosophy in Mathematics. This dissertation has not been submitted for the award of any degree, diploma, associateship, fellowship or other title. I hereby confirm the originality of the work and that there is no plagia- rism in any part of the dissertation. Place: Bengaluru Date: Dr Mayamma Joseph Professor Department of Mathematics Christ University, Bengaluru. Dr T V Joseph Head of the Department Department of Mathematics Christ University, Bengaluru. iv ACKNOWLEDGMENT First and foremost, I thank God with reverence and honour who blessed me good health, courage, ingenuity, zeal and zest to fulfill my dreams. I would like to express my sincere and deepest gratitude to my adorable guide Dr Mayamma Joseph for being patient enough to plough through preliminary versions of my text to elevate my performance. I appreciate all her contributions of time, expertise, invaluable guidance, affec- tionate attitude, understanding, patience and healthy criticism. Without her constant inspiration, it would have not been possible to accomplish this thesis. I am extremely grateful to Dr (Fr) Abraham V M, The Pro-Vice Chancellor; Dr T V Joseph, Head of the Department of Mathematics and all faculty members of the department of mathe- matics for providing me an opportunity to become a part of Christ University. I express my sincere thanks to Dr (Fr) Joseph Varghese and Dr Mukti Acharya for their motivational and facilitative nature. I would like to acknowledge my friends Annet and Meenakshy for their extended support. I learnt a lot from them through their personal and scholarly interactions. I owe a lot to my parents, Shripad and Hema, who motivated and was always there for me in every situation of my life. I could reach this stage because of them. I am highly thankful to my siblings, specially my sister Vrushali for her emotional support throughout my life. My heartfelt regards goes to my father-in-law Balkrishna and my mother-in-law Bhagyashri for their affection and considerateness. I am greatly indebted to a very special person, my husband Bharat for his continuous and unfailing love, understanding and patience during my M.phil. programme. Completion of this thesis would have not been possible without his support. He always stood by my decisions and boosted my confidence when I thought to give up. I appreciate my little angel Saee for her patience which contributed immensely to complete my thesis. At last I thank one and all who have been a part of the process of completion of this thesis directly or indirectly. Prajakta Bharat Joshi v ABSTRACT In this dissertation, we extend the study of color energy of graphs initiated by Adiga et al. We obtain some new bounds for the color energy of graphs and the upper bound in terms of the largest positive color eigenvalue l1 is a better bound than the upper bound for color energy in [4]. Also we establish relationship between color energy Ec(G) and energy E(G) of a graph G. Moreover, we prove some relations of color energy with Zagreb index, Laplacian energy and signless Laplacian energy. Further, We construct two new families of graphs and derive explicit formulas for their color energy with respect to the minimum number of colors c(G). Also we determine few families of graphs which are non-co-spectral color-equienergetic and few families of graphs which are color-hyperenergetic. We compute exact values of color energy of paths and fan graphs upto order 50. vi Contents Approval of dissertation ii Declaration iii Certificate iv Acknowledgment v Abstract vi Contents vii List of figures x List of tables xii 1 Introduction 1 1.1 Background . 1 1.2 Preliminaries . 2 1.3 Graph coloring . 11 vii 1.3.1 Vertex coloring . 11 1.3.2 Edge coloring . 13 1.4 Graph energy . 14 1.4.1 Chemical graphs . 15 1.4.2 Energy of graphs . 17 1.5 Review of literature . 19 1.6 Outline of the dissertation . 25 2 Color energy of a graph 26 2.1 Introduction . 26 2.2 Bounds for color energy of graphs . 28 2.2.1 Bounds for color energy in terms of l1 and lmax . 28 0 2.2.2 Bounds for color energy in terms of m and mc . 31 2.3 Relation between color energy and energy of a graph . 33 2.4 Bounds for color energy in terms of a topological index and other energies . 34 2.4.1 Bounds for color energy in terms of Zagreb index . 35 2.4.2 Bounds for color energy in terms of Laplacian energy and signless Laplacian energy . 38 3 Non-co-spectral color-equienergetic and color-hyperenergetic graphs 41 3.1 Non-co-spectral color-equienergetic graphs . 41 3.2 Color-hyperenergetic graphs . 45 3.2.1 Color energies of paths and fan graphs upto order 50 . 47 viii Bibliography 52 ix List of Figures 1.1 A graph G ..................................... 2 1.2 Two isomorphic graphs H1 and H2 ........................ 3 1.3 A graph G2 with a pendent and an isolated vertex . 3 1.4 Paths and cycles . 4 1.5 Complete graphs of order upto 5 . 4 1.6 Bipartite and complete bipartite graphs . 5 1.7 A star graph K1;5 ................................. 5 1.8 Petersen graph . 5 1.9 A wheel graph W6 ................................ 6 1.10 A fan graph F6 ................................... 6 1.11 A chordal graph and a non-chordal graph . 6 1.12 A graph F and its various subgraphs . 7 1.13 A graph G3 .................................... 8 1.14 A graph G, its complement and some self complementary graphs . 8 1.15 A graph G4 .................................... 9 1.16 A graph F1 .................................... 9 x 1.17 Connected and disconnected graphs . 10 1.18 Proper vertex coloring of Petersen graph . 10 1.19 A model graph G with proper vertex coloring for the storage problem . 12 1.20 The street intersection with six traffic lanes . 12 1.21 The model graph G and a 3-coloring of G .................... 13 1.22 A model graph of the scheduling problem . 14 1.23 Molecular structures of propane and cyclopropane . 15 1.24 Molecular graphs of propane and cyclopropane . 15 1.25 Molecular structures of cyclobutane and mythylenecyclopropene . 16 1.26 Hydrogen suppressed molecular graphs of cyclobutane and mythylenecyclo- propene . 16 1.27 Benzocyclobutadiene . 17 1.28 Kekulé structure and Kekulé graph of Benzocyclobutadiene . 17 1.29 Two examples of conjugated hydrocarbon . 18 1.30 Molecular graphs of the compounds in the Figure 1.29 . 18 2.1 P4 ......................................... 27 2.2 Graph G ...................................... 30 3.1 S = K1;n−1 + e .................................. 42 3.2 H = S + e .................................... 45 3.3 Comparison of energy of paths and its color energy . 49 3.4 Comparison of energy of fan graphs and its color energy . 49 xi List of Tables 1.1 Color energy of some graphs and their color compliments . 21 3.1 Energy and color energy of Pn and Fn, for n ≤ 50 . 48 xii Chapter 1 Introduction 1.1 Background Graph theory is the study of mathematical discrete structures. It is gaining importance in re- search due to its relevance in several disciplines such as physics, chemistry, computer science, psychology and social science. Some of the major topics of interest in graph theory are alge- braic graph theory, domination, decomposition, extremal graph theory, labeling, graph coloring, graph energy etc. Graph theory is a powerful tool which is used to prove some fundamental results in pure math- ematics like Farmat’s little therem and Nielson-Schreier theorem. Recently, some new appli- cations of graph theory are developed for DNA sequencing and computer network security. In DNA sequencing, SNP (Single Nucleotide Polymorphism) assembly problem is a problem for removing few possible SNP’s so that all the conflicts are eliminated where SNP is a single base mutation in DNA and worm propagation problem in computer network security is a problem for protecting computer networks from virus attacks in polynomial time. These problems can be solved by using vertex cover algorithm [24]. Another interesting example of applications of graph theory can be page rank algorithm.
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