31,394 research outputs found
An Edge-Swap Heuristic for Finding Dense Spanning Trees
Finding spanning trees under various restrictions has been an interesting question to researchers. A dense tree, from a graph theoretical point of view, has small total distances between vertices and large number of substructures. In this note, the density of a spanning tree is conveniently measured by the weight of a tree (defined as the sum of products of adjacent vertex degrees). By utilizing established conditions and relations between trees with the minimum total distance or maximum number of sub-trees, an edge-swap heuristic for generating dense spanning trees is presented. Computational results are presented for randomly generated graphs and specific examples from applications
Structural Properties of Planar Graphs of Urban Street Patterns
Recent theoretical and empirical studies have focused on the structural
properties of complex relational networks in social, biological and
technological systems. Here we study the basic properties of twenty
1-square-mile samples of street patterns of different world cities. Samples are
represented by spatial (planar) graphs, i.e. valued graphs defined by metric
rather than topologic distance and where street intersections are turned into
nodes and streets into edges. We study the distribution of nodes in the
2-dimensional plane. We then evaluate the local properties of the graphs by
measuring the meshedness coefficient and counting short cycles (of three, four
and five edges), and the global properties by measuring global efficiency and
cost. As normalization graphs, we consider both minimal spanning trees (MST)
and greedy triangulations (GT) induced by the same spatial distribution of
nodes. The results indicate that most of the cities have evolved into networks
as efficienct as GT, although their cost is closer to the one of a tree. An
analysis based on relative efficiency and cost is able to characterize
different classes of cities.Comment: 7 pages, 3 figures, 3 table
Graph measures and network robustness
Network robustness research aims at finding a measure to quantify network
robustness. Once such a measure has been established, we will be able to
compare networks, to improve existing networks and to design new networks that
are able to continue to perform well when it is subject to failures or attacks.
In this paper we survey a large amount of robustness measures on simple,
undirected and unweighted graphs, in order to offer a tool for network
administrators to evaluate and improve the robustness of their network. The
measures discussed in this paper are based on the concepts of connectivity
(including reliability polynomials), distance, betweenness and clustering. Some
other measures are notions from spectral graph theory, more precisely, they are
functions of the Laplacian eigenvalues. In addition to surveying these graph
measures, the paper also contains a discussion of their functionality as a
measure for topological network robustness
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