14,986 research outputs found
Kemeny's constant and Wiener index on trees
On trees of fixed order, we show a direct relation between Kemeny's constant
and Wiener index, and provide a new formula of Kemeny's constant from the
relation with a combinatorial interpretation. Moreover, the relation simplifies
proofs of several known results for extremal trees in terms of Kemeny's
constant for random walks on trees. Finally, we provide various families of
co-Kemeny's mates, which are two non-isomorphic connected graphs with the same
Kemeny's constant, and we also give a necessary condition for a tree to attain
maximum Kemeny's constant for trees with fixed diameter
The analysis of increasing trees and other families of trees
9502325T
Faculty of Science
School of MathematicsAbstract
Increasing trees are labelled rooted trees in which labels along any branch from the root appear in increasing order. They have numerous applications in tree representations of permutations, data structures in computer science and probabilistic models in a multitude of problems. We use a generating function approach for the computation of parameters arising from such trees. The generating functions for some parameters are shown to be related to ordinary differential equations. Singularity analysis is then used to analyze several parameters of the trees asymptotically.Various classes of trees are considered. Parameters such as depth and path length for heap ordered trees have been analyzed in [35]. We follow a similar approach to determine grand averages for such trees. The model is that p of the n nodes are labelled at random in ôn
p(ways, and the characteristic parameters depend on these labelled nodes. Also, we will
attempt to look at the limiting distributions involved. Often, when they are Gaussian, Hwang's quasi power theorem, from [18], can be employed. Spanning tree size and the Wiener index for binary search trees have been computed in [33]. The Wiener index is the sum of all distances between pairs of nodes in a tree. Arelated parameter of interest is the Steiner distance which generalises, to sets of k nodes, the Wiener index (k=2). Furthermore, the distribution of the size of the ancestor-tree and of the induced spanning subtree for random trees is presented in [30]. The same procedure is followed to obtain the Steiner distance for heap ordered trees and for other varieties of increasing trees
On weighted depths in random binary search trees
Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg.
Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled
rooted tree is the sum of all labels on the path connecting the node to the
root. We analyze weighted depths of nodes with given labels, the last inserted
node, nodes ordered as visited by the depth first search process, the weighted
path length and the weighted Wiener index in a random binary search tree. We
establish three regimes of nodes depending on whether the second order
behaviour of their weighted depths follows from fluctuations of the keys on the
path, the depth of the nodes, or both. Finally, we investigate a random
distribution function on the unit interval arising as scaling limit for
weighted depths of nodes with at most one child
The Minimum Wiener Connector
The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph and a set
of query vertices, find a subgraph of that connects all
query vertices and has minimum Wiener index.
We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless . Our main contribution is a
constant-factor approximation algorithm running in time
.
A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set a small number of important vertices
(i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International
Conference on Management of Dat
The center of mass of the ISE and the Wiener index of trees
We derive the distribution of the center of mass of the integrated
superBrownian excursion (ISE) {from} the asymptotic distribution of the Wiener
index for simple trees. Equivalently, this is the distribution of the integral
of a Brownian snake. A recursion formula for the moments and asymptotics for
moments and tail probabilities are derived.Comment: 11 page
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