Approximating the Inertia of the Adjacency Matrix of a Connected Planar Graph That Is the Dual of a Geographic Surface Partitioning Daniel A. Griffith, Ulemu Luhanga School of Economic, Political and Policy Sciences, University of Texas at Dallas, Richardson, TX 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. Introduction Geography has a long research tradition involving graph theory (e.g., Kansky 1963; Haggett and Chorley 1969). Part of this earlier work embraces eigenfunctions of graphs (e.g., Tinkler 1972; Boots 1984; Boots and Royle 1991; Mac ´kiewicz and Ratajczak 1996). More recent work pertains to spatial weights matrices employed in spatial statistics and spatial econometrics (Cliff and Ord 1981; Tiefelsdorf and Boots 1995; Griffith 2003). A considerable part of this more recent work involves the full set of eigenvalues for spatial weights matrices, many of which are not known analytically and hence have to be calculated, a numerically challenging task when the number of locations involved becomes very large to massive. Correspondence: Daniel A. Griffith, School of EPPS, University ofTexas at Dallas, 800 W. Campbell Road, Richardson, TX 75080 e-mail: dagriffith@utdallas.edu Daniel A. Griffith is an Ashbel Smith Professor of Geospatial Information Sciences; Ulemu Luhanga is a doctoral student in the Geospatial Information Sciences Program. Submitted: July 1, 2010. Revised version accepted: July 5, 2011. Geographical Analysis ISSN 0016-7363 Geographical Analysis 43 (2011) 383–402 © 2011 The Ohio State University 383