On K-trees and Special Classes of K-trees

The class of k-trees is defined recursively as follows: the smallest k-tree is the k-clique. If G is a graph obtained by attaching a vertex v to a k-clique in a k-tree, then G is also a k-tree. Trees, connected acyclic graphs, are k-trees for k = 1. We introduce a new parameter known as the shell of a k-tree, and from the shell special subclasses of k-trees, tree-like k-trees, are classified. Tree-like k-trees are generalizations of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. Let fs = fs( G) be the number of independent sets of cardinality s of G. Then the polynomial I(G; x) = [special characters omitted] fs(G)x s is called the independence polynomial. 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. Let M1 and M2 be the first and second Zagreb index respectively. Then 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. Additionally, the Zagreb indices of tree-like k-trees are investigated

On Topological Indices And Domination Numbers Of Graphs

Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well