We introduce novel matrices for graphs embedded on twoand three-dimensional grids. The matrices are defined in terms of geometrical and topological distances in such graphs. We report on some properties of these distance/distance matrices and have listed several structural invariants derived from distance/distance matrices. The normalized Perron root (the first eigenvalue) of such matrices, \/n, for path graphs apparently is an index of molecular folding. The ratio = / is 1 for (geometrically) linear structures, while it approaches 0 as the path graph is repeatedly folded.Graphs have found considerable use in chemistry,1 par-

Several necessary or sufficient conditions for a sign pattern to allow eventual positivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.

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We consider rigorous definitions for dense graphs and sparse matrices, thus quantifying these concepts that have been hitherto used in a qualitative manner. We assign to every graph the compactness index F, which is a measure of density of graphs and binary matrices. This index takes values F > 1 for dense graphs and F < 1 for sparse graphs (matrices). The index is derived from the quotient of the relative graph density and the percentage of zero entries in the adjacency matrix. The numerical values for the compactness index are reported for miscellaneous graphs, including several families of structurally related graphs.

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any selfconjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [5], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured that the converse is true as well; we present counterexamples to both of these statements. In [2] it was conjectured that any n × n spectrally arbitrary sign pattern must have at least 2n nonzero entries; we establish that this conjecture is true for 5 × 5 sign patterns. We also establish analogous results for nonzero patterns.