<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper describes the previously unknown statistical distribution of adjacency matrix spectra for planar graphs, also known as spatial weights matrices, in terms of the following three readily available eigenvalue properties: extremes, rank orderings, and sums of powers. This distribution is governed by at most six parameters that, once known, allow accurate approximations of eigenvalues to be computed without resorting to numerical matrix methods applied on a case-by-case basis. Parameter estimates for illustrative real-world examples are obtained using nonlinear least squares regression techniques. Three conjectures are proposed, and graphical and trend results are reported for a diverse set of planar graph-based matrices.</p></abstract>

This paper describes the previously unknown statistical distribution of adjacency matrix spectra for planar graphs, also known as spatial weights matrices, in terms of the following three readily available eigenvalue properties: extremes, rank orderings, and sums of powers. This distribution is governed by at most six parameters that, once known, allow accurate approximations of eigenvalues to be computed without resorting to numerical matrix methods applied on a case-by-case basis. Parameter estimates for illustrative real-world examples are obtained using nonlinear least squares regression techniques. Three conjectures are proposed, and graphical and trend results are reported for a diverse set of planar graph-based matrices.

On The Eigenvalue Distribution Of Adjacency Matrices For Connected Planar Graphs

School of Economic, Political, and Policy Sciences, University of Texas at Dallas, USA

Quaestiones Geographicae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On The Eigenvalue Distribution Of Adjacency Matrices For Connected Planar Graphs"}

This paper describes the previously unknown statistical distribution of adjacency matrix spectra for planar graphs, also known as spatial weights matrices, in...

On The Eigenvalue Distribution Of Adjacency Matrices For Connected Planar Graphs

Published Online: Dec 30, 2015

Page range: 39 - 60

Received: Feb 02, 2015

DOI: https://doi.org/10.1515/quageo-2015-0035

Keywords
adjacency matrix, connected graph, eigenvalue distribution, planar graph, serial structure

© 2015 Faculty of Geographical and Geological Sciences, Adam Mickiewicz University

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On The Eigenvalue Distribution Of Adjacency Matrices For Connected Planar Graphs

Published Online: Dec 30, 2015

Page range: 39 - 60

Received: Feb 02, 2015

DOI: https://doi.org/10.1515/quageo-2015-0035

Keywordsadjacency matrix, connected graph, eigenvalue distribution, planar graph, serial structure

© 2015 Faculty of Geographical and Geological Sciences, Adam Mickiewicz University

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
adjacency matrix, connected graph, eigenvalue distribution, planar graph, serial structure