Papers by Richard A Brualdi
Nested species subsets, gaps, and discrepancy
Oecologia, 1999
The nested-subset hypothesis of Patterson and Atmar states that species composition on islands wi... more The nested-subset hypothesis of Patterson and Atmar states that species composition on islands with less species richness is a proper subset of those on islands with greater species richness. The sum of species absences, referred to as gaps, was suggested as a metric for nestedness, and null models have been used to test whether or not island species exhibited nestedness. Simberloff and Martin stated that finding examples of non-nested faunas was difficult. We revisit previous analyses of nested faunas and introduce a new metric we call “discrepancy” which we recommend as a measure for nestedness. We also recommend that the sample spaces conserve both row sums (number of species per site) and column sums (number of sites per species) derived from the incidence matrix. We compare our results to previous analyses.
An interesting face of the polytope of doubly stochastic matrices
Linear & Multilinear Algebra, 1985
We consider a face of dimension (n − n)/2 of the polytope of n × n doubly stochastic matrices. We... more We consider a face of dimension (n − n)/2 of the polytope of n × n doubly stochastic matrices. We determine the minimum permanent on this face and show that the matrices in which achieve this minimum permanent form a convex polytope whose dimension is asymptotic to the dimension of Several problems and conjectures are proposed.
Journal of Combinatorial Theory, 1977
Received March 21, I975
Journal of Combinatorial Theory, 1991
The assignment polytope
Mathematical Programming, 1976
Journal of Combinatorial Theory, 1977
Received March 21, I975
Journal of Combinatorial Theory, 1977
Received March 21, I975
Linear Algebra and Its Applications, 1980
Let m and n be positive integers, and let R = (rl,. . . , r,) and S = (an.. . ,sn) be nonnegative... more Let m and n be positive integers, and let R = (rl,. . . , r,) and S = (an.. . ,sn) be nonnegative integral vectors. We survey the combinatorial properties of the set of all m x n matrices of O's and l's having ri l's in row i and si I's in column i. A number of new results are proved. The results can also be formulated in terms of the set of bipartite graphs with a bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S. of Fu(R,S) . 1 d' g mc u m some recent results. In doing so, we present in some cases new proofs of theorems which may be more transparent than those in the literature. Also there appear here for the first time a number of new results, notably the solution (Theorem 6.8) of a problem posed by Ryser [56, p. 761 in 1963. Other new results include Theorems 3.10, 4.2, 4.4, 5.8, 5.9, 6.8, 6.10, 7.3, 8.3, and 8.13, and Corollaries 5.6 and 8.6. Over twenty problems are proposed. Before proceeding we give two alternative interpretations of 8(R, S). Let X=(x,,..., xm} and Y={ yr,..., y,} be disjoint sets of m and n elements, respectively. Let BG(R,S) d enote the collection of all bipartite graphs G with the following properties: (BGl) The vertices of G are xi,. . . , x,, yl,. . . , y,,. (BG2) Each edge of G joins a vertex in X to a vertex in Y. (BG3) The degree (or valency) of xi is r, for i = 1,. . . , m, and the degree of yj is si for j=l,...,n. Then there is a one-to-one correspondence between the matrices in '%(R,S) and the bipartite graphs in BG(R, S), determined as follows. If A =[a,] E
Discrete Mathematics, 1993
We define the incidence coloring number of a graph and bound it in terms of the maximum degree. T... more We define the incidence coloring number of a graph and bound it in terms of the maximum degree. The incidence coloring number turns out to be the strong chromatic index of an associated bipartite graph. We improve a bound for the strong chromatic index of bipartite graphs all of whose cycle lengths are divisible by 4.
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Papers by Richard A Brualdi