A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,−1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, −1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in simil...

Let U(R, S) denote the class of all m x n matrices of O's and l's having row sum vector R and column sum vector S. The interchange graph G (R, S) is the graph where the vertices are the matrices in U (R, S) and two vertices are adjacent provided they differ by an interchange. Two tight upper-bounds of the maximum degree l:l.(G(R, S)) are given. Furthermore, those extreme graphs whose maximum de...

Position-weight matrices (PWMs) are broadly used to locate transcription factor binding sites in DNA sequences. The majority of existing PWMs provide a low level of both sensitivity and specificity. We present a new computational algorithm, a modification of the Staden-Bucher approach, that improves the PWM. We applied the proposed technique on the PWM of the GC-box, binding site for Sp1. The c...

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n) := 1!4!7! · · · (3n − 2)! n!(n + 1)! · · · (2n − 1)! . Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) ‘1’ of the first row is at the rth column equals A(n) `n+r−2 n−1 ́`2n−1−r n−1 ́ `3n−2 n−1 ́ . Standing on...

Here we solve N × N Riemann-Hilbert (inverse monodromy) problems with all monodromy matrices having the structure of matrices of quasi-permutation (i.e. matrices which have only one non-zero element in each column and each row). Such RiemannHilbert problem may be associated to arbitrary Hurwitz space of algebraic curves L of genus g realized as N -sheeted covering over CP1, and allowes solution...

We prove an asymptotic estimate for the number of m×n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of m × n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization proble...

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices; that is, the set of real non-negative matrices with all row and column sums equal to one. We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of “magic squares”, i.e., n×n non-negative integer matrices whose rows and columns a...