Approximating the Inertia of the Adjacency Matrix of a Connected Planar Graph That Is the Dual of a Geographic Surface Partitioning

This article addresses the calculation of the inertia of an adjacency matrix (i.e., the number of positive, zero, and negative eigenvalues) associated with a connected, undirected planar graph. A formula is derived that is an upper bound for the number of negative eigenvalues of this matrix, based upon standard matrix trace results, coupled with the use of nonextreme eigenvalue averages, and requiring calculations of the pair of extreme eigenvalues and the number of zero eigenvalues. The number of positive eigenvalues can be calculated easily from this specific result. Assessment of this formula is in terms of selected regular two‐dimensional tessellations and in terms of a set of empirical surface partitioning, commonly employed in spatial analyses. Proposed correction factors allow a modification of this formula to estimate more precisely the associated inertia of an adjacency matrix. Este artículo aborda el cálculo de la inercia de una matriz de adyacencia, (es decir, el número de valores propios (eigenvalues) positivos, cero y negativos) asociados a un grafo conexo plano no orientado (connected undirected planar graph). Se deriva una fórmula que es el límite superior del número de valores propios negativos de esta matriz basándose en los resultados estándar de la traza de la matriz (matrix trace) junto con el uso de los promedios de valores propios no extremos y que requiere cálculos del par de valores propios extremos y del número de valores propios cero. El número de valores propios positivos se puede calcular fácilmente a partir de este resultado. Se evalúa esta fórmula en términos de mosaicos regulares bidimensionales y en términos de un conjunto de superficies de partición (surface partitioning) empíricas comúnmente empleados en análisis espaciales. Los factores de corrección propuestos permiten modificar esta fórmula para estimar con mayor precisión la inercia de una matriz de adyacencia. 本文讨论了联通非有向平面图邻接矩阵的惯性逼近（即正的,零以及负的特征值的个数）问题。基于标准矩阵迹的结果,综合利用非极值特征值的平均值,通过计算极值特征值对和零特征值个数,构建了邻接矩阵中负特征值上界计算公式。正特征值的个数可以简单地从上述结果中直接计算。采用空间分析中常用的规则二维格网化选取,以及一系列的经验曲面剖分对该公式的评估。所提出的修正指数允许对该公式进行修正,从而更精确地估计邻接矩阵惯性。     

The Majority Theorem for the Single (p = 1) Median Problem and Local Spatial Autocorrelation

Except for about a half dozen papers, virtually all (co)authored by Griffith, the existing literature lacks much content about the interface between spatial optimization, a popular form of geographic analysis, and spatial autocorrelation, a fundamental property of georeferenced data. The popular p-median location-allocation problem highlights this situation: the empirical geographic distribution of demand virtually always exhibits positive spatial autocorrelation. This property of geospatial data offers additional overlooked information for solving such spatial optimization problems when it actually relates to their solutions. With a proof-of-concept outlook, this paper articulates connections between the well-known Majority Theorem of the 1-median minisum problem and local indices of spatial autocorrelation; the LISA statistics appear to be the more useful of these later statistics because they better embrace negative spatial autocorrelation. The relationship articulation outlined here results in the positing of a new proposition labeled the egalitarian theorem.