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INDEPENDENCE POLYNOMIALS OF CIRCULANT GRAPHS by Richard Hoshino SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY AT DALHOUSIE UNIVERSITY HALIFAX, NOVA SCOTIA OCTOBER 15, 2007 c Copyright by Richard Hoshino, 2007 DALHOUSIE UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled \Independence Polynomials of Circulant Graphs" by Richard Hoshino in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dated: October 15, 2007 External Examiner: Dr. Michael Plummer Research Supervisor: Dr. Jason Brown Examining Committee: Dr. Jeannette Janssen Dr. Richard Nowakowski ii DALHOUSIE UNIVERSITY Date: October 15, 2007 Author: Richard Hoshino Title: Independence Polynomials of Circulant Graphs Department: Mathematics and Statistics Degree: Ph.D. Convocation: May Year: 2008 Permission is herewith granted to Dalhousie University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission. The author attests that permission has been obtained for the use of any copyrighted material appearing in the thesis (other than brief excerpts requiring only proper acknowledgement in scholarly writing) and that all such use is clearly acknowledged. iii This is the true joy in life, the being used for a purpose recognized by yourself as a mighty one, the being a force of nature, instead of a selfish, feverish little clod of ailments and grievances complaining that the world will not devote itself to making you happy. I am of the opinion that my life belongs to the whole community, and it is my privilege to do for it whatever I can. Life is no brief candle to me, it is a sort of splendid torch which I've got a hold of for the moment, and I want to make it burn as brightly as possible before handing it on to future generations. { George Bernard Shaw iv Table of Contents List of Tables vii List of Figures viii Abstract ix List of Symbols Used x Acknowledgements xi Chapter 1 Introduction 1 1.1 Basic Terminology . 1 1.2 Circulant Graphs . 2 1.3 The Independence Polynomial . 4 1.4 Overview of the Thesis . 7 Chapter 2 Formulas for Independence Polynomials 9 2.1 The Family S = 1; 2; : : : ; d . 9 f g 2.2 The Family S = d + 1; d + 2; : : : ; n . 15 f b 2 cg 2.3 Circulants of Degree at Most 3 . 31 2.4 Graph Products of Circulants . 40 2.5 Evaluating the Independence Polynomial at x = t . 49 2.6 Independence Unique Graphs . 54 Chapter 3 Analysis of a Recursive Family of Circulant Graphs 63 3.1 Calculating the Independence Number α(Cn;S) . 63 3.2 Application 1: Star Extremal Graphs . 73 3.3 Application 2: Integer Distance Graphs . 107 3.4 Application 3: Fractional Ramsey Numbers . 117 3.5 Application 4: Nordhaus-Gaddum Inequalities . 125 v Chapter 4 Properties and Applications of Circulant Graphs 144 4.1 Line Graphs of Circulants . 144 4.2 List Colourings . 158 4.3 Well-Covered Circulants . 171 4.4 Independence Complexes of Circulant Graphs . 200 Chapter 5 Roots of Independence Polynomials 220 5.1 The Real Roots of I(Cn;S; x) . 222 5.2 The Root of Minimum Modulus of I(Cn;S; x) . 226 5.3 The Root of Maximum Modulus of I(Cn;S; x) . 236 5.4 The Rational Roots of I(Cn;S; x) . 240 5.5 The Roots of Independence Polynomials and their Closures . 247 Chapter 6 Conclusion 254 Bibliography 258 vi List of Tables Table 3.1 The nine known Ramsey numbers of the form r(a; b). 119 Table 4.1 Eulerian subgraphs for the circulant C9; 1;2 . 161 f g Table 4.2 Connected well-covered circulants on at most 12 vertices. 172 Table 4.3 Connected well-covered circulants with G 12 and α(G) 3. 219 j j ≤ ≥ Table 4.4 Shelling of the pure complex ∆(C12; 1;3;6 ). 219 f g Table 5.1 Examples of graphs for which I(Bn; x) is not stable. 225 Table 5.2 The closures of the roots of graph polynomials. 253 vii List of Figures Figure 1.1 The circulant graphs C9; 1;2 and C9; 3;4 . 3 f g f g Figure 1.2 Two independent sets of size 3 in C6. 4 Figure 2.1 The ladder graph Un. 32 Figure 2.2 The graphs G9 = C18; 1;9 and H9 = C18; 2;9 . 32 f g f g Figure 2.3 The graphs G = C18; 2;9 and H = C18; 8;9 . 35 f g f g Figure 2.4 The graph products C 2C and C C . 41 3 3 3 × 3 Figure 2.5 Two trees with the same independence polynomial. 55 Figure 2.6 The graphs G = C8; 3;4 and the corresponding graph H3. 57 f g 5 Figure 3.1 A 3-colouring of C5 and a fractional 2 -colouring of C5. 74 Figure 4.1 Possible subgraphs of G induced by these 7 edges. 149 Figure 4.2 Possible subgraphs of G induced by e0; e1; e2; e3; en 1 . 150 f − g Figure 4.3 Two possible edge labellings of K4. 153 Figure 4.4 Two possible K subgraphs induced by e ; e ; e ; e ; e ; e : 155 4 f 0 a 2a 3a 4a 5ag Figure 4.5 Two possible subgraphs induced by this set of eight edges. 157 Figure 4.6 A 2-list colouring of K2;4 that is not proper. 159 Figure 4.7 A 3-list colouring of K7;7 that is not proper. 160 Figure 4.8 Graphs which generate the family of cubic graphs. 177 Figure 4.9 Four well-covered cubic graphs. 178 Figure 4.10 Graph X and its vertices labelled. 179 Figure 4.11 Graph Y and its vertices labelled. 180 Figure 4.12 Graph Z and its vertices labelled. 181 Figure 4.13 Separating nk = aknk 1 1 vertices into ak classes. 189 − − Figure 4.14 The independence complex ∆(C6). 202 Figure 4.15 The independence complex ∆(3K2). 202 viii Abstract The circulant graph C is the graph on n vertices (with labels 0; 1; 2; : : : ; n 1), n;S − spread around a circle, where two vertices u and v are adjacent iff their (minimum) distance u v appears in set S. In this thesis, we provide a comprehensive analysis of j − j the independence polynomial I(G; x), when G = Cn;S is a circulant. The independence polynomial is the generating function of the number of independent sets of G with k vertices, for k 0. ≥ While it is NP -hard to determine the independence polynomial I(G; x) of an arbitrary graph G, we are able to determine explicit formulas for I(G; x) for several families of circulants, using techniques from combinatorial enumeration. We then describe a recursive construction for an infinite family of circulants, and determine the independence number of each graph in this family. We use these results to provide four applications, encompassing diverse areas of discrete mathematics. First, we determine a new (infinite) family of star extremal graphs. Secondly, we obtain a formula for the chromatic number of infinitely many integer distance graphs. Thirdly, we prove an explicit formula for the generalized fractional Ramsey function. Finally, we determine the optimal Nordhaus-Gaddum inequalities for the fractional chromatic and circular chromatic numbers. These new theorems significantly extend what is currently known. Building on these results, we develop additional properties and applications of circulant graphs. We determine a full characterization of all graphs G for which its line graph L(G) is a circulant, and apply our previous theorems to calculate the list colouring number of a specific family of circulants. We then characterize well-covered circulant graphs, and examine the shellability of their independence complexes. We conclude the thesis with a detailed analysis of the roots of I(Cn;S; x). Among many other results, we solve several open problems by calculating the density of the roots of these independence polynomials, leading to new theorems on the roots of matching polynomials and rook polynomials. ix Abstract Cn;S The circulant graph on n vertices with generating set S I(G; x) The independence polynomial of graph G Pn The path on n vertices Cn The cycle on n vertices Kn The complete graph on n vertices G The order of a graph G, i.e., the number of vertices in G j j u v The circular distance of two vertices u and v in C j − jn n;S deg(v) The degree of a vertex v α(G) The independence number of G !(G) The clique number of a G χ(G) The chromatic number of G G The complement of G L(G) The line graph of G [xk]P (x) The coefficient of the xk term of P (x) An The circulant Cn; 1;2;:::;d , where d 1 is fixed f g ≥ Bn The circulant Cn; d+1;d+2;:::; n , where d 0 is fixed f b 2 cg ≥ G[H] The lexicographic graph product of G with H χf (G) The fractional chromatic number of G χc(G) The circular chromatic number of G !f (G) The fractional clique number of G χl(G) The list colouring number of G G(Z; S) The integer distance graph generated by set S ρ(G) The vertex arboricity number of G r(a1; a2; : : : ; ak) The Ramsey function of the k-tuple (a1; a2; : : : ; ak) r!f (a1; a2; : : : ; ak) The fractional Ramsey function of the k-tuple (a1; a2; : : : ; ak) ∆(G) The independence complex of G x Acknowledgements First and foremost, I'd like to thank my supervisor Jason Brown, for his incredible support and patience during the completion of this thesis.
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