On K-Trees and Special Classes of K-Trees

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

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 independence 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) = u∈V (G) d(u) , and M2(G) = uv∈E(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<