The Terminal Wiener Index of Trees with Diameter or Maximum Degree

The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arises in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents a sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees which attain these bounds. In addition, we investigate the properties of extremal trees which attain the maximum terminal Wiener index among all trees of order $n$ with fixed maximum degree.Comment: 15 pages, 1 figur

Generalizations of Wiener polarity index and terminal Wiener index

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertices of degree $k$. For $k = 1$, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order $n$.Comment: 3pages, 4 figure

On Minimum Terminal Distance Spectral Radius of Trees with Given Degree Sequence

For a tree with the given sequence of vertex degrees the spectral radius of its terminal distance matrix is shown to be bounded from below by the the average row sum of the terminal distance matrix of the, so called, BFS-tree (also known as a greedy tree). This lower bound is typically not tight; nevertheless, since spectral radius of the terminal distance matrix of BFS-tree is a natural upper bound, the numeric simulation shows that relative gap between the upper and the lower bound does not exceed 3% (we also make a step towards justifying this fact analytically.) Therefore, the conjecture that BFS-tree has the minimum terminal distance spectral radius among all trees with the given degree sequence is valid with accuracy at least 97%. The same technique can be applied to the distance spectral radius of trees, which is a more popular topological index.Comment: 21 pages, 5 figure

Tree statistics from Matula numbers

There is a one-to-one correspondence between natural numbers and rooted trees; the number is called the Matula number of the rooted tree. We show how a large number of properties of trees can be obtained directly from the corresponding Matula number

On the average Steiner 3-eccentricity of trees

The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V(G)$ which contain $v$. In this paper Steiner $3$-eccentricity is studied on trees. Some general properties of the Steiner $3$-eccentricity of trees are given. A tree transformation which does not increase the average Steiner $3$-eccentricity is given. As its application, several lower and upper bounds for the average Steiner $3$-eccentricity of trees are derived

Mathematical aspects of Wiener index

The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.Comment: 28 pages, 4 figures, survey pape

Lower bound for the cost of connecting tree with given vertex degree sequence

The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, structure identification from data, etc. For the case of connecting trees with the given sequence of vertex degrees, the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life data sets. Keywords: Optimal communication network, generalized Wiener index, origin-destination matrix, semidefinite programming, quadratic matrix inequality.Comment: 29 pages, 6 figures, 2 table

Some useful lemmas on the edge Szeged index

The edge Szeged index of a graph $G$ is defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. In this paper, we characterize the graph with minimum edge Szeged index among all the unicyclic graphs with given order and diameter

Enumerating the total number of subtrees of trees

Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number of subtrees (resp. leaf containing subtrees) among $n$-vertex trees with a given matching number is determined; as a consequence, the $n$-vertex tree with domination number $\gamma$ maximizing the total number of subtrees (resp. leaf containing subtrees) is characterized. (2)\, Sharp lower bound on the total number of leaf containing subtrees among $n$-vertex trees with maximum degree at least $\Delta$ is determined; as a consequence the $n$-vertex tree with maximum degree at least $\Delta$ having a perfect matching minimizing the total number of subtrees (resp. leaf containing subtrees) is characterized. (3)\, Sharp upper (resp. lower) bound on the total number of leaf containing subtrees among the set of all $n$-vertex trees with $k$ leaves (resp. the set of all $n$-vertex trees of diameter $d$) is determined.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1204.615

Some Aspects of the Wiener Index for Sun Graphs

The Wiener index $W(G)$ is the sum of distances of all pairs of vertices of the graph $G$. The Wiener polarity index $W_{p}(G)$ of a graph $G$ is the number of unordered pairs of vertices $u$ and $v$ of $G$ such that the distance $d_{G}(u,v)$ between $u$ and $v$ is $3$. In this paper the Wiener and the Wiener polarity indices of sun graphs are computed. A relationship between those indices with some other topological indices are presented. Finally, we find the Hosoya (Wiener) polynomial for sun graphs