Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-19

December 2000

Binet matrices, an extension of network matrices

Gautam Appa and Balázs Kotnyek


We define a new class of matrices called binet matrices - B, closely related to network matrices - N. We show that if A belongs to class B, then the polyhedron P(Ab) = { x : Ax <= b,  x >= 0 } is integral for all integral b vectors whose common divisor is 2. Starting with the node-edge incidence matrices of bidirected graphs having elements 0, 1 and 2, binet matrices with elements 0, 1, 2 and 1/2 are derived as their generalization, in parallel with the derivation of network matrices as generalised node-edge incidence matrices of directed graphs. It is shown that the two well known totally unimodular matrices which are not network matrices are binet matrices, but the transpose of a binet matrix is not necessarily a binet matrix. The most striking result obtained is that class B is closed under pivoting and taking submatrices. Some further theorems on binet matrices are developed, leaving open the problem of an efficient recognition algorithm.

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