DOCSLIB.ORG

Energy of Graphs and Digraphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Energy of Graphs and Digraphs University of Lethbridge Research Repository OPUS http://opus.uleth.ca Theses Arts and Science, Faculty of 2010 Energy of graphs and digraphs Jahanbakht, Nafiseh Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science, c2010 http://hdl.handle.net/10133/2489 Downloaded from University of Lethbridge Research Repository, OPUS ENERGY OF GRAPHS AND DIGRAPHS NAFISEH JAHANBAKHT Mathematics, Shahid Beheshti University, 2002 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Nafiseh Jahanbakht, 2010 This thesis is dedicated to Dr. Kourosh Tavakoli. iii Abstract The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept is related to the energy of a class of molecules in chemistry and was first brought to mathematics by Gutman in 1978 ([8]). In this thesis, we do a comprehensive study on the energy of graphs and digraphs. In Chapter 3, we review some existing upper and lower bounds for the energy of a graph. We come up with some new results in this chapter. A graph with n vertices is hyper-energetic if its energy is greater than 2n 2. Some classes of graphs are proved to be hyper-energetic. We find − a new class of hyper-energetic graphs which is introduced and proved to be hyper-energetic in Section 3.3. The energy of a digraph is the sum of the absolute values of the real part of the eigenvalues of its adjacency matrix. In Chapter 4, we study the energy of digraphs in a way that Pe˜na and Rada in [19] have defined. Some known upper and lower bounds for the energy of digraphs are re- viewed. In Section 4.5, we bring examples of some classes of digraphs in which we find their energy. Keywords. Energy of a graph, hyper-energetic graph, energy of a di- graph. iv Acknowledgments I am deeply indebted to my advisor, Dr. Hadi Kharaghani, whose com- passionate supervision was invaluable. It was a great honor for me to work under his supervision. I would also like to thank Dr. Wolf Holzmann, Dr. Amir Akbary, and Dr. Mark Walton for their helps and supports. I would like to thank Lily Liu for her new ideas that helped me write Section 3.3. I would like to warmly thank my father who was my first teacher and showed me the joy of learning, and my mother for her endless kind- ness. They have always been my endless source of inspiration. I would also like to thank my spouse, Kourosh Tavakoli, to whom this thesis is dedicated, for all his encouragements and supports at the time I was com- pletely disappointed. I would like to thank my sister, Nahal, whose recovery is the best gift from God. The last but not the least is my brother, Mohammad Mehdi, whom I would like to deeply thank for all his supports from the first day I started my program at the University of Lethbridge. v Contents Approval/Signature Page ii Dedication iii Abstract iv Acknowledgments v Table of Contents vi List of Figures vii 1 Introduction 1 2 Preliminaries 8 2.1 BasicDefinitions ..................... 8 2.2 Eigenvaluesofagraph . 11 2.3 StronglyRegularGraphs . 18 3 Energy of Graphs 22 3.1 Minimalenergygraphs . 26 3.2 MaximalEnergyGraphs . 29 3.3 Hyper-energeticGraphs. 37 3.3.1 GeneralizationandMainResult . 48 4 Energy of Digraphs 55 4.1 Introduction........................ 55 4.2 CoefficientsTheorem . 55 4.3 Integralrepresentationoftheenergy . 60 4.4 Upperandlowerbounds . 63 4.4.1 Upperboundfortheenergyofdigraphs . 63 4.4.2 Lowerboundfortheenergyofdigraphs . 69 4.5 Energyofsomedigraphs . 71 Bibliography 78 vi List of Figures 1.1 G2 ............................. 3 1.2 C4 ............................. 5 2.1 Pseudoregulargraphs. 10 2.2 Paleygraphoforder13 . 18 3.1 Contour Γ ......................... 23 3.2 Ahyper-energeticgraphon8vertices . 38 3.3 G2 ............................. 42 3.4 Anexampleoflocalstronglyregulargraphs . 49 4.1 Digraph........................... 58 4.2 Contour Γ ......................... 61 4.3 P7 : Paleydigraphoforder7 . 72 4.4 Digraphs for matrices D1 and D2 ............. 74 vii Chapter 1 Introduction The concept of the energy of a graph was introduced three decades ago by Ivan Gutman [8]. This notion is related to the total electron energy of a class of organic molecules in computational chemistry. The total energy of the so-called π-electrons is calculated by the formula n Eπ = ∑ λ j (1.1) j=1 | | where n is the number of the molecular orbital energy levels and λ js are eigenvalues of the adjacency matrix of the so-called molecular or H¨uckel graph. Although in chemistry the expression (1.1) is valid only for the class of “H¨uckel graphs”, the right-hand side of (1.1) is well-defined for any class of graphs in mathematics. This motivated Gutman to define the energy of a graph. Definition 1.0.1 [8] Let G be a graph, the energy of G, denoted by E(G), is the sum of the absolute values of the eigenvalues of G, i.e. if λ1,...,λn n are the eigenvalues of G, then E(G) = ∑ λi . i=1 | | For one example, by using the eigenvalues found in page 14, the energy of a complete graph of order n is computed as E(Kn) = n 1 + 1 (n 1) = 2n 2 . | − | | − | − − 1 There are some bounds on the energy of a graph. In this thesis, we mention some of the most well-known bounds. For a graph G of order n with m edges, McClelland ([17]) in early 70’s, gave the following general bounds on its energy where A is the adjacency matrix of G. 2 2m + n(n 1) det(A) n E(G) √2mn. − | | ≤ ≤ q A lower bound for the energy of a graph only in terms of its number of edges is E(G) 2√m with equality if and only if G is a complete ≥ bipartite graph plus some isolated vertices. In terms of the number of vertices the lower bound is E(G) 2√n 1 with equality if and only G ≥ − is the star K1,n 1. − For the upper bound, there is a well-known result due to Koolen and Moulton ([14]) which is an improvement on the McCelland bound. For a graph G with n vertices and m edges where 2m n, they proved ≥ 2m 2m 2 E(G) + (n 1) 2m n v n ≤ u − " − # u t n with equality if and only if G is Kn, 2 K2, or a strongly regular graph (2m (2m/n)2) (SRG) with two eigenvalues having absolute value −(n 1) . − Next, if we consider the left hand side of the aboveq inequality as a function of m, it is maximized when m =(n2 + n√n)/4. By substituting 2 this amount in the above formula we find n(1 + √n) E(G) . (1.2) ≤ 2 Koolen and Moulton ([14]) proved that (1.2) is also valid for 2m < n and that the equality holds if and only if G is an SRG with parameters (n,(n + √n)/2,(n + 2√n)/4,(n + 2√n)/4). They also conjectured that for a given ε > 0 there exists a graph G of n order n such that for almost all n 1, E(G) (1 ε) (√n + 1) which ≥ ≥ − 2 was later proved in [18] by Nikiforov. There was a conjecture in 1978 that between graphs of order n, the complete graph Kn has the maximum energy. Although it was rejected and it was shown that there exist subgraphs of Kn with energy greater than that of Kn, it was an introduction for defining hyper-energetic graphs. A graph G with n vertices is hyper-energetic if E(G) > 2n 2. Some − classes of graphs have shown to be hyper-energetic. Figure 1.1: G2 In this thesis, we introduce a new class of graphs which we prove that they are hyper-energetic. Our first example in this class is a 4-regular 3 graph with 13 vertices that we call G2 (see Figure 1.1). In general case, we construct Gm as follows. Consider 2m 2 copies − of K2m and m copies of K2 and one copy of K1. Add edges to make it 2m-regular by adding 2m edges from the single vertex K1 to vertices of m copies of K2. Then add one edge from each vertex of K2m to the m copies of K2 (see Figure 3.4 in Section 3.3.1). Gm isa2m-regular graph with n =(2m 1)2m+1 vertices. We found − 2m 3 that the characteristic polynomial of Gm is (x 2m)(x (2m 1)) − − − − (x + 1)(2m 3)(2m 1)(x2 (2m 1))m(x2 + 2x (2m 3))m and from that − − − − − − we can find the energy of Gm. Then we prove that for m > 2, graph Gm is hyper-energetic. If we want to generalize the concept of energy for the case of di- graphs, we should be reminded that the adjacency matrix is not symmet- ric and the eigenvalues might be complex numbers. Pe˜na and Rada in [19] proposed the following definition for the energy of digraphs. Definition 1.0.2 Let G be a digraph, the energy of G, denoted by E(G), is the sum of the absolute values of the real part of the eigenvalues of G. In fact, Pe˜na and Rada proved the Coulson integral formula for the case of digraphs ([19]) and that was the motivation for the Definition 1.0.2.
Load more

Recommended publications
  • Discrete Mathematics Lecture Notes Incomplete Preliminary Version
    Discrete Mathematics Lecture Notes Incomplete Preliminary Version Instructor: L´aszl´oBabai Last revision: June 22, 2003 Last update: April 2, 2021 Copyright c 2003, 2021 by L´aszl´oBabai All rights reserved. Contents 1 Logic 1 1.1 Quantifier notation . 1 1.2 Problems . 1 2 Asymptotic Notation 5 2.1 Limit of sequence . 6 2.2 Asymptotic Equality and Inequality . 6 2.3 Little-oh notation . 9 2.4 Big-Oh, Omega, Theta notation (O, Ω, Θ) . 9 2.5 Prime Numbers . 11 2.6 Partitions . 11 2.7 Problems . 12 3 Convex Functions and Jensen's Inequality 15 4 Basic Number Theory 19 4.1 Introductory Problems: g.c.d., congruences, multiplicative inverse, Chinese Re- mainder Theorem, Fermat's Little Theorem . 19 4.2 Gcd, congruences . 24 4.3 Arithmetic Functions . 26 4.4 Prime Numbers . 30 4.5 Quadratic Residues . 33 4.6 Lattices and diophantine approximation . 34 iii iv CONTENTS 5 Counting 37 5.1 Binomial coefficients . 37 5.2 Recurrences, generating functions . 40 6 Graphs and Digraphs 43 6.1 Graph Theory Terminology . 43 6.1.1 Planarity . 52 6.1.2 Ramsey Theory . 56 6.2 Digraph Terminology . 57 6.2.1 Paradoxical tournaments, quadratic residues . 61 7 Finite Probability Spaces 63 7.1 Finite probability space, events . 63 7.2 Conditional probability, probability of causes . 65 7.3 Independence, positive and negative correlation of a pair of events . 67 7.4 Independence of multiple events . 68 7.5 Random graphs: The Erd}os{R´enyi model . 71 7.6 Asymptotic evaluation of sequences .
    [Show full text]
  • Octonion Multiplication and Heawood's
    CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7].
    [Show full text]
  • PDF of the Phd Thesis
    Durham E-Theses Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs THOMAS, DANIEL,JAMES,RHYS How to cite: THOMAS, DANIEL,JAMES,RHYS (2020) Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13671/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas A Thesis presented for the degree of Doctor of Philosophy Department of Computer Science Durham University United Kingdom June 2020 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas Submitted for the degree of Doctor of Philosophy June 2020 Abstract: In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdős, Ko and Rado to graphs.
    [Show full text]
  • A Paley-Like Graph in Characteristic Two Andrew Thomason
    Journal of Combinatorics Volume 7, Number 2–3, 365–374, 2016 A Paley-like graph in characteristic two Andrew Thomason To Adrian Bondy on his seventieth birthday The Paley graph is a well-known self-complementary pseudo-ran- dom graph, defined over a finite field of odd order. We describe an attempt at an analogous construction using fields of even order. Some properties of the graph are noted, such as the existence of a Hamiltonian decomposition. 1. Introduction The well-known Paley graph is a pseudo-random graph whose vertex set is the finite field Fq = GF (q)oforderq ≡ 1 (mod 4). The pair of vertices a, b is joined by an edge if a − b is a square in Fq.Since−1isasquarethe graph is well defined. It follows from elementary properties of the quadratic character that the Paley graph is vertex-transitive, self-complementary, and each edge is in (q − 5)/4 triangles. A graph of order q in which every edge is in (q − 5)/4 triangles and whose complement has the same property is sometimes called a conference graph. The Paley graph is thus ipso facto a pseudo-random graph, as explained in detail in [12],andinasomewhatless quantitative fashion in Chung, Graham and Wilson in [3]. The other odd prime powers, namely those where q ≡ 3 (mod 4), cannot be used to construct Paley graphs since −1 is not a square. However this very property allows the construction of a tournament, or oriented complete graph, on the vertex Fq by inserting an edge oriented from a to b if a − b is a square.
    [Show full text]
  • Arxiv:1702.00285V1 [Math.HO] 31 Jan 2017 Paley and the Paley Graphs
    Paley and the Paley graphs Gareth A. Jones Abstract This paper discusses some aspects of the history of the Paley graphs and their automorphism groups. MSC Classifications: 01A60, 05-03, 05B05, 05B20, 05E30, 11E25, 12E20, 20B25, 51M20 1 Introduction Anyone who seriously studies algebraic graph theory or finite permutation groups will, sooner or later, come across the Paley graphs and their automorphism groups. The most frequently cited sources for these are respectively Paley’s 1933 paper [67], and Carlitz’s 1960 paper [14]. It is remarkable that neither of those papers uses the concepts of graphs, groups or automorphisms. Indeed, one cannot find these three terms, or any synonyms for them, in those papers: Paley’s paper is entirely about the construction of what are now called Hadamard matrices, while Carlitz’s is entirely about permutations of finite fields. The aim of the present paper is to explore how this strange situation came about, by explaining the background to these two papers and how they became associated with the Paley graphs. This involves describing various links with other branches of mathematics, such as matrix theory, number theory, design theory, coding theory, finite geometry, poly- tope theory and group theory. The paper is organised in two main parts, the first covering arXiv:1702.00285v1 [math.HO] 31 Jan 2017 the graphs and the second their automorphism groups, each largely in historical order. However, in order to establish basic concepts we start with the definition and elementary properties of the Paley graphs. 2 Definition and properties of the Paley graphs e The Paley graph P (q) has vertex set V = F := Fq, a field of prime power order q = p ≡ 1 mod (4), with two vertices u and v adjacent if and only if u − v is an element of the set S = {x2 | x ∈ F, x =06 } of quadratic residues (non-zero squares) in F.
    [Show full text]
  • Codes from Incidence Matrices and Line Graphs of Paley Graphs
    Advances in Mathematics of Communications doi:10.3934/amc.2011.5.93 Volume 5, No. 1, 2011, 93{108 CODES FROM INCIDENCE MATRICES AND LINE GRAPHS OF PALEY GRAPHS Dina Ghinelli Dipartimento di Matematica Universit`adi Roma `La Sapienza' I-00185 Rome, Italy Jennifer D. Key School of Mathematical Sciences University of KwaZulu-Natal Pietermaritzburg 3209, South Africa (Communicated by Ivan Landjev) Abstract. We examine the p-ary codes from incidence matrices of Paley graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes q(q−1) q−1 q(q−1) q−1 are [ 4 ; q − 1; 2 ]2 or [ 4 ; q; 2 ]p for p odd. By finding PD-sets we show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q). 1. Introduction The codes from the row span of adjacency matrices of Paley graphs are the well- known quadratic residue codes and studied in many texts. We look here at the codes from the row span of incidence matrices of Paley graphs, following the ideas of some earlier papers that studied such codes for various classes of graphs and their line graphs, including Hamming graphs, complete graphs, and complete bipartite graphs: see [14, 16,6,7,8]. An incidence matrix G for a graph Γ = (V; E) is a jV j × jEj matrix with rows labelled by the vertices and columns by the edges and an entry 1 if a vertex is on edge, 0 otherwise.
    [Show full text]
  • Spectra of Graphs
    Spectra of graphs Andries E. Brouwer Willem H. Haemers 2 Contents 1 Graph spectrum 11 1.1 Matricesassociatedtoagraph . 11 1.2 Thespectrumofagraph ...................... 12 1.2.1 Characteristicpolynomial . 13 1.3 Thespectrumofanundirectedgraph . 13 1.3.1 Regulargraphs ........................ 13 1.3.2 Complements ......................... 14 1.3.3 Walks ............................. 14 1.3.4 Diameter ........................... 14 1.3.5 Spanningtrees ........................ 15 1.3.6 Bipartitegraphs ....................... 16 1.3.7 Connectedness ........................ 16 1.4 Spectrumofsomegraphs . 17 1.4.1 Thecompletegraph . 17 1.4.2 Thecompletebipartitegraph . 17 1.4.3 Thecycle ........................... 18 1.4.4 Thepath ........................... 18 1.4.5 Linegraphs .......................... 18 1.4.6 Cartesianproducts . 19 1.4.7 Kronecker products and bipartite double. 19 1.4.8 Strongproducts ....................... 19 1.4.9 Cayleygraphs......................... 20 1.5 Decompositions............................ 20 1.5.1 Decomposing K10 intoPetersengraphs . 20 1.5.2 Decomposing Kn into complete bipartite graphs . 20 1.6 Automorphisms ........................... 21 1.7 Algebraicconnectivity . 22 1.8 Cospectralgraphs .......................... 22 1.8.1 The4-cube .......................... 23 1.8.2 Seidelswitching. 23 1.8.3 Godsil-McKayswitching. 24 1.8.4 Reconstruction ........................ 24 1.9 Verysmallgraphs .......................... 24 1.10 Exercises ............................... 25 3 4 CONTENTS 2 Linear algebra 29 2.1
    [Show full text]
  • Shannon Capacity and the Categorical Product
    Shannon capacity and the categorical product G´abor Simonyi∗ Alfr´edR´enyi Institute of Mathematics, Budapest, Hungary and Department of Computer Science and Information Theory, Budapest University of Technology and Economics [email protected] Submitted: Oct 31, 2019; Accepted: Feb 16, 2021; Published: Mar 12, 2021 © The author. Released under the CC BY-ND license (International 4.0). Abstract Shannon OR-capacity COR(G) of a graph G, that is the traditionally more of- ten used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore COR(F × G) 6 minfCOR(F );COR(G)g holds for every pair of graphs, where F × G is the categorical product of graphs F and G. Here we initiate the study of the question when could we expect equality in this in- equality. Using a strong recent result of Zuiddam, we show that if this \Hedetniemi- type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much \nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on COR(F × G) and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound minfCOR(F );COR(G)g. We present such graph pairs using the properties of Paley graphs.
    [Show full text]
  • Graph Theory – CMSC-27500 – Spring 2015 Homework Set #6. Posted 4-17, 2Am, Up
    Graph Theory { CMSC-27500 { Spring 2015 http://people.cs.uchicago.edu/∼laci/15graphs Homework set #6. Posted 4-17, 2am, updated 6:30am Problem 6.20 added at 4:40pm, 6.21 at 11:45pm. Due Tuesday, April 21, typeset in LaTeX. Do not submit homework before its due date; it may get lost by the time we need to grade them. If you must submit early, write the early submissions on separate sheets, separately stapled; state \EARLY SUBMIS- SION" on the top, and send email to the instructor listing the problems you submitted early and the reason of early submission. Read the homework instructions on the website. The instructions that follow here are only an incomplete summary. Hand in your solutions to problems marked \HW" and \BONUS." Do not hand in problems marked \DO." Warning: the BONUS problems are underrated. PRINT YOUR NAME ON EVERY SHEET you submit. Use LaTeX to typeset your solutions. (You may draw diagrams by hand.) Hand in your solutions on paper, do not email. If you hand in solutions to CHALLENGE problems, do so on a separate sheet, clearly marked \CHAL- LENGE," and notify the instructor by email to make sure it won't be over- looked. Carefully study the policy (stated on the website) on collaboration, in- ternet use, and academic integrity. State collaborations and sources both in your paper and in email to the instructor. Definitions, notation. As before G = (V; E) denotes a graph with n vertices and m edges. Notation: for a non-negative integer n we write [n] = f1; : : : ; ng.
    [Show full text]
  • Paley Graphs and Their Generalizations
    Paley Graphs and Their Generalizations By Ahmed Noubi Elsawy A thesis submitted to Heinrich Heine University D¨usseldorf, Germany for the Degree of Master of Science Supervisors Prof. Dr. F. Grunewald Prof. Dr. E. Klimenko arXiv:1203.1818v1 [math.CO] 7 Mar 2012 Department of Mathematics Heinrich Heine University D¨usseldorf 2009 MASTER OF SCIENCE THESIS OF Ahmed Elsawy APPROVED: THESIS COMMITTEE: MAJOR PROFESSOR DEAN OF THE GRADUATE SCHOOL HEINRICH HEINE UNIVERSITY DUSSELDORF¨ 2009 Acknowledgment I would like to offer my deepest gratitude and thankfulness to Prof. Dr. F. Grunewald for suggesting the topics of this thesis, for his kind supervision, and for his invaluable help during the progress of present work. I owe my deepest gratitude and special thankfulness to Prof. Dr. E. Klimenko who offered me a great help in understanding the topics and to get this document into its present shape. Many thanks for her stimulat- ing discussion, sincere advice, incisive comments, and careful reading of the manuscript. Naturally, all the mistakes that can still be found in this thesis are mine. If you ever find a word that is spelled correctly, it must have been Prof. Dr. E. Klimenko dropping me a line. I want also to thank my family for their support and for keeping me going. A. Elsawy Contents Introduction 7 1 Finite Fields 11 1.1 Basic definitions and properties . 11 1.2 Classification of the finite fields . 14 1.3 Constructionofthefinitefields . 18 2 Paley Graph 23 2.1 Basic definitions and properties . 23 2.2 Paleygraphs............................ 27 2.2.1 Definition and examples .
    [Show full text]
  • Paley Graphs and Paley Tournaments
    Paley graphs and Paley tournaments L´aszl´oBabai Last updated 2021 May 21 at 0:20 1 Congruences Z denotes the set of integers. Definition 1.1. Let a; b 2 Z. We say that a divides b (notation: a j b) if (9x)(ax = b). (Here x is also an integer.) Exercise 1.2. Note that 0 j 0. Definition 1.3. Let a; b; m 2 Z. We say that a is congruent to b modulo m if m j a − b. Notation: a ≡ b (mod m). Exercise 1.4. Let a ≡ b (mod m). Prove: gcd(a; m) = gcd(b; m). Exercise 1.5. Prove: for a fixed m, the relation a ≡ b (mod m) is an equivalence relation on Z. Terminology 1.6. The equivalence classes of this relation are called mod m residue classes. Exercise 1.7. Observe: a ≡ b (mod 0) if and only if a = b. Notation 1.8. Let [i]m denote the residue class of i mod m. We call i a representative of this residue class. Exercise 1.9. [i]m = [j]m if and only if i ≡ j (mod m). Exercise 1.10. Let m 6= 0. Then the number of mod m residue classes is jmj. Exercise 1.11. gcd([i]m; m) := gcd(i; m) is well-defined (it does not depend on the particular choice of representative i). Definition 1.12. [Arithmetic of residue classes] Let p be a prime number. We define [a]m + [b]m := [a + b]m and [a]m · [b]m := [ab]m.
    [Show full text]
  • Cayley Graphs. Journal of Combinatorial Theory, Series B, Pages 180–189, 1979
    Spectral Graph Theory Lecture 5 Cayley Graphs Daniel A. Spielman September 12, 2018 5.1 Cayley Graphs Ring graphs and hypercubes are types of Cayley graph. In general, the vertices of a Cayley graph are the elements of some group Γ. In the case of the ring, the group is the set of integers modulo n. The edges of a Cayley graph are specified by a set S ⊂ Γ, which are called the generators of the Cayley graph. The set of generators must be closed under inverse. That is, if s 2 S, then s−1 2 S. Vertices u; v 2 Γ are connected by an edge if there is an s 2 S such that u ◦ s = v; where ◦ is the group operation. In the case of Abelian groups, like the integers modulo n, this would usually be written u + s = v. The generators of the ring graph are f1; −1g. d The d-dimensional hypercube, Hd, is a Cayley graph over the additive group (Z=2Z) : that is the set of vectors in f0; 1gd under addition modulo 2. The generators are given by the vectors in f0; 1gd that have a 1 in exactly one position. This set is closed under inverse, because every element of this group is its own inverse. We require S to be closed under inverse so that the graph is undirected: u + s = v () v + (−s) = u: Cayley graphs over Abeliean groups are particularly convenient because we can find an orthonormal basis of eigenvectors without knowing the set of generators. They just depend on the group1.
    [Show full text]