Pseudo-centrosymmetric matrices, with applications to counting perfect matchings

Pseudo-centrosymmetric matrices, with applications to counting perfect matchings

We consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K2 = I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K2 = −I, we define A to be pseudo-centrosymmetric with respect to K; we show that the determinant of every even-order pseudo-centrosymmetric matrix is the sum of two square...

Perfect Matchings and Applications

On Counting Perfect Matchings in General Graphs

Counting perfect matchings has played a central role in the theory of counting problems. The permanent, corresponding to bipartite graphs, was shown to be #P-complete to compute exactly by Valiant (1979), and a fully polynomial randomized approximation scheme (FPRAS) was presented by Jerrum, Sinclair, and Vigoda (2004) using a Markov chain Monte Carlo (MCMC) approach. However, it has remained a...

Counting perfect matchings and the switch chain

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chai...

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least (n + 1)!/4[q − p − (n − 1)(2 − 3) + 4] different perfect matchings, where is the maximum degree of a vertex in G. © 2007 Elsevier B.V. All rights reserved. MSC: 05C70; 05C40; 05C75