3,339 research outputs found
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
On the Randi\'{c} index and conditional parameters of a graph
The aim of this paper is to study some parameters of simple graphs related
with the degree of the vertices. So, our main tool is the matrix
whose ()-entry is where denotes the degree of the vertex . We study
the Randi\'{c} index and some interesting particular cases of conditional
excess, conditional Wiener index, and conditional diameter. In particular,
using the matrix or its eigenvalues, we obtain tight bounds on the
studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243
Cacti with Extremal PI Index
The vertex PI index is a
distance-based molecular structure descriptor, where denotes the
number of vertices which are closer to the vertex than to the vertex
and which has been the considerable research in computational chemistry dating
back to Harold Wiener in 1947. A connected graph is a cactus if any two of its
cycles have at most one common vertex. In this paper, we completely determine
the extremal graphs with the largest and smallest vertex PI indices among all
the cacti. As a consequence, we obtain the sharp bounds with corresponding
extremal cacti and extend a known result.Comment: Accepted by Transactions on Combinatorics, 201
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