416 research outputs found
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
Loopy, Hankel, and Combinatorially Skew-Hankel Tournaments
We investigate tournaments with a specified score vector having additional
structure: loopy tournaments in which loops are allowed, Hankel tournaments
which are tournaments symmetric about the Hankel diagonal (the anti-diagonal),
and combinatorially skew-Hankel tournaments which are skew-symmetric about the
Hankel diagonal. In each case, we obtain necessary and sufficient conditions
for existence, algorithms for construction, and switches which allow one to
move from any tournament of its type to any other, always staying within the
defined type
A generalization of Alternating Sign Matrices
In alternating sign matrices the first and last nonzero entry in each row and
column is specified to be +1.
Such matrices always exist. We investigate a generalization by specifying
independently the sign of the first and last nonzero entry in each row and
column to be either a +1 or a -1. We determine necessary and sufficient
conditions for such matrices to exist.Comment: 14 page
Review of recent results in the theory of graph spectra by D. Cvetković, M. Doob, I. Gutman, and A. Torĝasev
On the t-Term Rank of a Matrix
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to
be the largest number of 1s in A with at most one 1 in each column and at most
t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize
some basic results for the term rank to the t-term rank, including a formula
for the maximum term rank over a nonempty class of (0,1)-matrices with the the
same row sum and column sum vectors. We also show the surprising result that in
such a class there exists a matrix which realizes all of the maximum terms
ranks between 1 and t.Comment: 18 page
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