The paper deals with the representation of a transportation infrastructure by a planar connected simple graph and aims at studying its features through the analysis of graph properties. All planar and connected graphs with 4 up to 7 edges are analysed and compared to extract the most suitable parameters to investigate some network features. Then, a set of 41 graphs representing some actual underground networks are also analysed. Besides, as a third scenario, the underground network of Milan, along its development in years, is proposed in order to apply the proposed methodology. Many parameters are taken into consideration. Some of them are already discussed in literature, such as the eigenvalues and gaps of adjacency matrix or such as the Bclassical^ parameters α, β, γ. Others, such as the first two Betti numbers, are new for these applications.In order to overcome the problem of comparing features of graphs with different size, the normalisation of these parameters is considered. Some relationships between Betti numbers, eigenvalues, and classical parameters are also investigated. Results show that the eigenvalues and gaps of the adjacency matrix well represent some features of the graphs while combining them with the Betti numbers, a more significant interpretation can be achieved. Particularly, their normalised values are able to describe the increasing complexity of a graph.

Structure Indicators for Transportation Graph Analysis I: Planar Connected Simple Graphs

MUSSONE, LORENZO;NOTARI, ROBERTO
2017

Abstract

The paper deals with the representation of a transportation infrastructure by a planar connected simple graph and aims at studying its features through the analysis of graph properties. All planar and connected graphs with 4 up to 7 edges are analysed and compared to extract the most suitable parameters to investigate some network features. Then, a set of 41 graphs representing some actual underground networks are also analysed. Besides, as a third scenario, the underground network of Milan, along its development in years, is proposed in order to apply the proposed methodology. Many parameters are taken into consideration. Some of them are already discussed in literature, such as the eigenvalues and gaps of adjacency matrix or such as the Bclassical^ parameters α, β, γ. Others, such as the first two Betti numbers, are new for these applications.In order to overcome the problem of comparing features of graphs with different size, the normalisation of these parameters is considered. Some relationships between Betti numbers, eigenvalues, and classical parameters are also investigated. Results show that the eigenvalues and gaps of the adjacency matrix well represent some features of the graphs while combining them with the Betti numbers, a more significant interpretation can be achieved. Particularly, their normalised values are able to describe the increasing complexity of a graph.
Transportation graph ,. Topological interpretation ,. Graph theory ,. Adjacency matrix , Eigenvalues , Betti numbers
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/975419
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