Higher order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition

Graph Invariants of Trees with Given Degree Sequence

Graph invariants are functions defined on the graph structures that stay the same under taking graph isomorphisms. Many such graph invariants, including some commonly used graph indices in Chemical Graph Theory, are defined on vertex degrees and distances between vertices. We explore generalizations of such graph indices and the corresponding extremal problems in trees. We will also briefly mention the applications of our results

Path-tables of trees: a survey and some new results

The (vertex) path-table of a tree $T$ contains quantitative information about the paths in $T$. The entry $(i,j)$ of this table gives the number of paths of length $j$ passing through vertex $v_i$. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree $T$ does not determine $T$ uniquely. We shall show that in trees, the number of paths passing through edge $xy$ can only be expressed in terms of paths passing through vertices $x$ and $y$ up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree $T$ of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables

Randic and Sum Connectivity Indices of Certain Trees

The focus of this thesis is on new development of the Randic and Sum Connec­tivity Indices of certain molecular and symmetric trees representing acyclic alkanes, or aliphatic hydrocarbons. The Randic Connectivity Index is one of the most used molecular descriptors in Quantitative Structure-Property and Structure-Activity Relationship modeling because of the relation that the isomers have between their properties and their structure. The structure-boiling point relationship models of aliphatic alcohols have been studied using the Sum Connectivity Index and compared to the Randic Connectivity Index. A specific type of tree Tn,a, well-known in graph theory as a double star, was studied by Zhou and Trinajstic. In this thesis, Tn,a trees are investigated. The tree Tn,2 which has the third smallest Sum Connectivity Index value among all the trees with n vertices is found to be interesting and thereby is further explored. Some alkane trees are symmetric, which is the concentration of this thesis. The symmetric double star trees are denoted by Jn· The tree Jn has n vertices and is built on the path P₂ with (n - 2)/2 leaves from each vertex of the path. The Randic and Sum Connectivity Index formulas of the symmetric tree ln are developed. Also, estimations of the Randic and Sum Connectivity Indices of Jn are given. Relationships and comparisons between the Randic and Sum Connectivity Indices are analyzed in respect to the tree Jn· The ratio and difference of the Randic and Sum Connectivity Indices are further discussed. The thesis starts with the history of the indices of molecular trees in Chemistry and Biology (Chapter 1). Chapter 2 provides a list of observations of the properties of both connectivity indices of the related trees. The symmetric tree Jn is discussed in Chapter 3, in which formulas and properties of the Randic and Sum Connectivity Indices are given. The main results of the thesis are reported in Chapter 4, where the graphs which have the maximal or minimal Randic and Sum Connectivity values among all Tn,a graphs with n vertices are identified. The closeness of the two indices of Tn,a trees is also discussed. The paper concludes with a similar tree, denoted Tn,a x Pm, extended from the tree Tn,a by replacing the middle path P2 with the path Pm (m \u3e/ 2). The Randic and Sum Connectivity Index formulas are given for this tree (Chapter 5). This topic will be investigated more in future work

Rearranging trees for robust consensus

In this paper, we use the H2 norm associated with a communication graph to characterize the robustness of consensus to noise. In particular, we restrict our attention to trees and by systematic attention to the effect of local changes in topology, we derive a partial ordering for undirected trees according to the H2 norm. Our approach for undirected trees provides a constructive method for deriving an ordering for directed trees. Further, our approach suggests a decentralized manner in which trees can be rearranged in order to improve their robustness.Comment: Submitted to CDC 201

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