Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs

Improved canopy reflectance modeling and scene inference through improved understanding of scene pattern

The Li-Strahler reflectance model, driven by LANDSAT Thematic Mapper (TM) data, provided regional estimates of tree size and density within 20 percent of sampled values in two bioclimatic zones in West Africa. This model exploits tree geometry in an inversion technique to predict average tree size and density from reflectance data using a few simple parameters measured in the field (spatial pattern, shape, and size distribution of trees) and in the imagery (spectral signatures of scene components). Trees are treated as simply shaped objects, and multispectral reflectance of a pixel is assumed to be related only to the proportions of tree crown, shadow, and understory in the pixel. These, in turn, are a direct function of the number and size of trees, the solar illumination angle, and the spectral signatures of crown, shadow and understory. Given the variance in reflectance from pixel to pixel within a homogeneous area of woodland, caused by the variation in the number and size of trees, the model can be inverted to give estimates of average tree size and density. Because the inversion is sensitive to correct determination of component signatures, predictions are not accurate for small areas

Marginally compact fractal trees with semiflexibility

We study marginally compact macromolecular trees that are created by means of two different fractal generators. In doing so, we assume Gaussian statistics for the vectors connecting nodes of the trees. Moreover, we introduce bond-bond correlations that make the trees locally semiflexible. The symmetry of the structures allows an iterative construction of full sets of eigenmodes (notwithstanding the additional interactions that are present due to semiflexibility constraints), enabling us to get physical insights about the trees' behavior and to consider larger structures. Due to the local stiffness the self-contact density gets drastically reduced.Comment: 16 pages, 12 figures, accepted for publication in PR