On [formula omitted]-span and [formula omitted]-span of trees and full binary trees

On subtrees of trees

The sum of distances between vertices of a tree has been considered from many aspects. The question of characterizing the extremal trees that maximize or minimize various such “distance-based” graph invariants has been extensively studied. Such invariants include, to name a few, the sum of distances between all pairs of vertices and the sum of distances between all pairs of leaves. With respect to the distances between internal vertices, we provide analogous results that characterize the extremal trees that minimize the value of any nonnegative and nondecreasing function of internal distances among trees with various constraints.

The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.

This paper considers partitioning the vertices of an n-vertex tree into p disjoint sets C1, C2, . . . , Cp, called clusters so that the number of vertices in a cluster and the number of subtrees in a cluster are minimized. For this NP-hard problem we present greedy heuristics which differ in (i) how subtrees are identified (using either a best-fit, good-fit, or first-fit selection criteria), (ii) whether clusters are filled one at a time or simultaneously, and (iii) how much cluster sizes can differ from the ideal size of c vertices per cluster, n = cp. The last criteria is controlled by a constant α, 0 ≤ α < 1, such that cluster Ci satisfies (1− α2 )c ≤ |Ci| ≤ c(1 + α), 1 ≤ i ≤ p. For algorithms resulting from combinations of these criteria we develop worst-case bounds on the number of subtrees in a cluster in terms of c, α, and the maximum degree of a vertex. We present experimental results which give insight into how parameters c, α, and the maximum degree of a vertex impact the number of subtrees and the cluster sizes. Communicated by G. Liotta: submitted November 1999, revised August 2000. 1. Hambrusch’s research supported in part by the National Science Foundation under Grant 9988339-CCR. 2. Lim’s research supported in part by Korea Science and Engineering Foundation under Contract No. 98-0102-07-01-3. S. E. Hambrusch et al., Clustering in Trees, JGAA, 4(4) 1–26 (2000) 2

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

The Wiener index is defined as the summation of distances between all pairs of vertices in a graph or in a hypergraph. Both models—graph-theoretical and hypergraph-theoretical—are used in mathematical chemistry for quantitatively studying physical and chemical properties of classical and nonclassical organic compounds. In this paper, we consider relationships between hypertrees and trees and hypercycles and cycles with respect to their Wiener indices.

The authors propose a VLSI layout strategy for a full binary tree. This layout can support more border leaf processing elements (PEs) and thus can give a higher I-O bandwidth. It is superior to the H-tree layout in terms of the number of boundary leaves. The approach uses H-tree pattern for constructing subtree layouts and then combines a number of such subtrees following standard tree style to get a larger sized tree layout. Finally at the top level H-tree layout style is used to get the overall tree layout. Different I/O bandwidths can be obtained varying the subtree height. The authors derive expression for layout area, longest link and aspect ratio of the chip. It is observed that I/O bandwidth can be significantly increased without much area overhead using this approach. >

The sum of distances between every pair of vertices in a graph G is called the Wiener index of G. This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of G does not account for any of its symmetries, which are also known to effect these physico-chemical properties. Graovac and Pisanski modified the Wiener index of G to measure the average distance each vertex is displaced under the elements of the symmetry group of G; we call this the Graovac-Pisanski (GP) distance number of G. In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group Γ and each rational number q within this half-line, we present a construction for a graph whose GP distance number is q and whose symmetry group is isomorphic to Γ. This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.

We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

The chemist Harold Wiener found 𝒲(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of 𝒲, called the Wiener polynomial 𝒲(G; q), is also useful, but it has few existing useful formulas. We will evaluate 𝒲(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

The distance d(u, v|G) between the vertices u and v of a simple connected graph G is the length of any shortest path in G connecting u and v. The Wiener index W(G) of G is defined as the sum of distances between all pairs of vertices of G. The edge-Wiener index of G is conceived in an analogous manner as the sum of distances between all pairs of edges of G. Two possible distances d 0(e, f|G) and d 4(e, f|G) between the edges e and f of G can be considered and according to them, the corresponding edge-Wiener indices \( {W}_{e_0}(G) \) and \( {W}_{e_4}(G) \) are defined. In this chapter, we report our recent results on computing the first and second edge-Wiener indices of some composite graphs. Results are illustrated by some interesting examples.

This paper studies embeddings of graphs in binary trees. The cost of such an embedding is the maximum distance in the binary tree between images of adjacent graph vertices. Several techniques for bounding the costs of such embeddings from above are derived; notable among these is an algorithm for embedding any outerplanar graph in a binary tree with a cost that is within a factor of 3 of optimal. A number of techniques for bounding the costs of such embeddings from below are developed; notable here are two techniques for inferring the presence of large separators in graphs. Finally, a number of characterizations are established of those families of graphs that are almost binary trees, in the sense that every graph in the family is embeddable in a binary tree within bounded cost.

Given an undirected graph G and two vertex subsets H 1 and H 2, the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H 1 and two edgedisjoint paths between every pair of vertices in H 2. We solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log2 n) time using n + m processors on an EREW PRAM.KeywordsCritical NodeGraph ProblemVertex SubsetConnectivity RequirementAugmentation ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Local properties of geometric graphs

Quasi-[formula omitted]-distance-balanced graphs