University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2012 On K-trees and Special Classes of K-trees John Wheless Estes Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Mathematics Commons Recommended Citation Estes, John Wheless, "On K-trees and Special Classes of K-trees" (2012). Electronic Theses and Dissertations. 101. https://egrove.olemiss.edu/etd/101 This Dissertation is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. ON k-TREES AND SPECIAL CLASSES OF k-TREES A Thesis presented for the Doctorate of Philosophy Department of Mathematics University of Mississippi JOHN WHELESS ESTES May 2012 Copyright John W. Estes 2011 ALL RIGHTS RESERVED ABSTRACT A tree is a connected graph with no cycles. In 1968 Beineke and Pippet introduced the class of generalized trees known as k-trees [3]. In this dissertation, we classify a subclass of k-trees known as tree-like k-trees and show that tree-like k-trees are a common generalization of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. A set I of vertices in a graph G is said to be independent if no pair of vertices of I are incident in G. Let fs = fs(G) be the number of independent sets of cardinality s of G. Pα(G) s Then the polynomial I(G; x) = s≥0 fs(G)x is called the independence polynomial of the graph G. [21]. In this dissertation, all rational roots of the independence polynomials of paths are found, and the exact paths whose independence polynomials have these roots are characterized. Additionally, trees are characterized that have −1=q as a root of their inde- pendence polynomials for 1 ≤ q ≤ 4. The well known vertex and edge reduction identities for independence polynomials are generalized, and the independence polynomials of k-trees are investigated. Additionally, sharp upper and lower bounds for fs of maximal outerplanar graphs, i.e. tree-like 2-trees, are shown along with characterizations of the unique maximal outerplanar graphs that obtain these bounds respectively. These results are extensions of the works of Wingard, Song et al., and Alameddine [1]. The first Zagreb index M1(G) and the second Zagreb index M2(G) of the graph G are P 2 P given by: M1(G) = u2V (G) d(u) , and M2(G) = uv2E(G) d(u)d(v). The study of the ii Zagreb indices M1 and M2 have been an active area of research since the report of Gutman and Trinajsti´c in computational chemistry [23] in 1972. The minimum and maximum M1 and M2 values for k-trees are determined, and the unique k-trees that obtain these minimum and maximum values respectively are characterized. In 2011, Hou, Li, Song, and Wei characterized the Zagreb indices for maximal outerplanar graphs and determined the unique maximal outerplanar graph that obtains minimum M1 and M2 values, respectively, as well as maximum M1 and M2 values respectively [29]. Select works of Hou et al. are extended to all tree-like k-trees. That is, the maximum M1 value for tree-like k-trees is determined, and the unique tree-like k-tree that obtains this maximum values respectively is characterized. Additionally, a partial result for the maximum M2 value for tree-like k-trees is determined, and a conjecture for a full result is presented. iii DEDICATION To Ethan, my son, who is my motivation to become a better person. iv LIST OF SYMBOLS G a graph V the vertex set of a graph G E the edge set of a graph G G[S] the subgraph induced by S G − v G[V − v] G − SG[V − S] G − e G remove an edge e G − FG remove a set of edges F G [ fuvg G add the edge uv G [ H the union of G and H jGj the cardinality of the vertex set of G jjGjj the cardinatlity of the edge set of G N(v) the neighborhood of vertex v N[v] the closed neighborhood of vertex v N(e) the neighborhood of edge e d(v) the degree of v δ(G) the minimum degree of G ∆(G) the maximum degree of G T a tree v d(v; u) the minimum distance from v to u in a graph Kn the complete graph on n vertices Pn the path on n vertices Sn the star on n vertices Kn1;n2 the complete bipartite graph on n1 + n2 vertices k Tn a k-tree on n vertices S1(G) the set of simplicial vertices in G k Pn the k-path on n vertices Sk;n−k the k-star on n vertices k Sn the k-spiral on n vertices k Dn the k-diamond on n vertices k Sh(Tn ) the shell of a k-tree τ(G) the toughness of G α(G) the independence number of G f(G) the fibonacci number of G fs(G) the number of indepedent sets of cardinality s of G I(G; x) the independence polynomial of G M1(G) the first Zagreb index M2(G) the second Zagreb index T (r) a p-descendant or p-ancestor of T vi β fTigi=0 a genealogy of a tree k k β(Tn ) the branching number of Tn vii ACKNOWLEDGEMENTS I thank my wife, Stephany, my source of constant support, encouragement, and inspiration. She is my partner who shares all of my stress, anxieties, laughter, and successes. I also thank my son, Ethan, for revealing to me a depth of love I had not previously known. Honor is due to my parents, Mickey and Suellen Estes, for all of the sacrifices they have made to invest into my future. I would not made it very far if it were not for their guidance, faith, and wisdom. I thank my three lovely sisters, Adrienne, Leslie, and Libby, and their families for all of the support and joyful memories. In addition, I would like to thank my wife's family, the Rangels, for their support as well. I thank Dr. Andrew Lang of Oral Roberts University for striving to mold a young and immature student into a mathematician. I also thank Dr. Bing Wei and Dr. William Staton of the University of Mississippi for the supervision of this dissertation. Their advice and counsel have been integral in developing the material presented here as well as my personal mathematical development. I thank Dr. James Reid, Dr. Haidong Wu, and Dr. Dawn Wilkins for taking time out of their busy schedules to serve on my dissertation committee. Their suggestions were well received. Additionally, I thank the faculty of the Department of Mathematics at the University of Mississippi for introducing many wonderful topics of mathematics to me and providing an environment of thought and growth. viii TABLE OF CONTENTS 1 INTRODUCTION . 1 2 TREE-LIKE k-TREES ............................................................ 9 3 INDEPENDENT SETS OF TREES . 19 4 INDEPENDENT SETS OF k-TREES . 31 5 THE ZAGREB INDICES OF k-TREES . 49 6 THE ZAGREB INDICES OF TREE-LIKE k-TREES . 74 7 TREE GENEALOGIES . 82 8 QUESTIONS FOR FUTURE RESEARCH . 96 LIST OF REFERENCES. 102 LIST OF APENDICES . 106 A TREES ON 1 ≤ n ≤ 10 VERTICES . 107 B INDEPENDENCE POLYNOMIALS OF TREES ON 1 ≤ n ≤ 10 VERTICES . 111 C THE ZAGREB INDICES OF TREE ON 1 ≤ n ≤ 10 VERTICES . 121 VITA................................................................................. 126 ix LIST OF FIGURES 1 A 4-tree on 10 vertices 6 2 The 3-path, 3-star, 3-spiral, and 3-diamond on 7 vertices 8 3 A 2-tree and its shell 9 4 T7;7 30 5 Maximal outerplanar graphs on n = 7 vertices 39 6 A tree-like 3-tree on 11 vertices G 41 7 Tree-like 3-trees on 8 vertices with τ > 1 45 8 A tree-like 4-tree on 10 vertices with toughness exceeding 1 46 9 The path and star on 7 vertices 50 10 The 2-path, 2-diamond, and 2-star on 6 vertices 75 11 The path and star on 7 vertices 82 12 A tree T and T (r) 83 13 A tree with no 1-ancestor 84 14 T5 89 15 T , T (r1), T (r2) 91 0 0 0 16 T , T (r1), T (r2) 92 x 17 A graph G and its shell 96 xi 1. Introduction A graph is chordal if it does not have an induced cycle of length greater than three. A graph is said to be k-degenerate if all of its subgraphs have minimum degree at most k, a concept introduced by Lick and White in 1970 [32], and a graph is maximally k-degenerate if it is k-degenerate and not a spanning subgraph of any other k-degenerate graph. In 1968 Beineke and Pippet introduced the class of generalized trees known as k-trees [3], and these graphs have attracted considerable research as well as many applications [4, 16, 33, 37, 38]. A purpose of this dissertation is to investigate the class of graphs that are both chordal and maximally k-degenerate, and it is shown that a graph is chordal and maximally k-degenerate if and only if it is a k-tree. A major emphasis of this dissertation is to classify a subclass of k-trees based on a \new" parameter known as the shell of a k-tree, which is a reformation of the (k + 1)-line graph first introduced in 2006 by Markenzon et al.