On the eigenvalue distribution of adjacency matrices for connected planar graphs

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2015

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Wydział Nauk Geograficznych i Geologicznych Uniwersytetu im. Adama Mickiewicza

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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.

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

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Quaestiones Geographicae vol. 34 (4), 2015, pp. 39-60

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Uniwersytet im. Adama Mickiewicza w Poznaniu
Biblioteka Uniwersytetu im. Adama Mickiewicza w Poznaniu
Ministerstwo Nauki i Szkolnictwa Wyższego