Result: Continuous mean distance of a weighted graph

Title:
Continuous mean distance of a weighted graph
Authors:
Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. DCCG - Discrete, Combinational, and Computational Geometry, Garijo Royo, Delia, Marquez Pérez, Alberto, Silveira, Rodrigo Ignacio
Publisher Information:
2023-08-01
Document Type:
Electronic Resource Electronic Resource
Index Terms:
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica, Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica aplicada a les ciències, Numerical analysis, Computer science--Mathematics, Weighted graph, Mean distance, Wiener index, Geometric graph, Algorithms, Anàlisi numèrica, Informàtica teòrica, Classificació AMS::65 Numerical analysis::65D Numerical approximation and computational geometry, Classificació AMS::68 Computer science::68R Discrete mathematics in relation to computer science, Article
URL:
http://hdl.handle.net/2117/391981
https://link.springer.com/article/10.1007/s00025-023-01902-w
Availability:
Open access content. Open access content
Attribution 4.0 International
https://creativecommons.org/licenses/by/4.0
Open Access
Note:
36 p.
application/pdf
English
Other Numbers:
HGF oai:upcommons.upc.edu:2117/391981
Garijo, D.; Marquez, A.; Silveira, R. Continuous mean distance of a weighted graph. "Results in Mathematics", 1 Agost 2023, vol. 78, núm. 139, p. 1-36.
1420-9012
10.1007/s00025-023-01902-w
1397547073
Contributing Source:
UNIV POLITECNICA DE CATALUNYA
From OAIster®, provided by the OCLC Cooperative.
Accession Number:
edsoai.on1397547073
Database:
OAIster

Further Information

The version of record of this article, first published in Results in Mathematics, is available online at Publisher’s website: http://dx.doi.org/10.1007/s00025-023-01902-w
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous analog of the (discrete) mean distance is the mean of the distances between all pairs of points on the edges of the graph. Despite being a very natural generalization, to the best of our knowledge this concept has been barely studied, since the jump from discrete to continuous implies having to deal with an infinite number of distances, something that increases the difficulty of the parameter. In this paper, we show that the continuous mean distance of a weighted graph can be computed in time roughly quadratic in the number of edges, by two different methods that apply fundamental concepts in discrete algorithms and computational geometry. We also present structural results that allow for a faster computation of this continuous parameter for several classes of weighted graphs. Finally, we study the relation between the (discrete) mean distance and its continuous counterpart, mainly focusing on the relevant question of convergence when iteratively subdividing the edges of the weighted graph.
Peer Reviewed
Postprint (published version)