Counting and selecting at random bipartite graphs with fixed degrees

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t. In particular, we find precise formulae for the probabilities that a given bipartite graph is edge-disjoint from, a subgraph of, or an induc...

Counting matchings in irregular bipartite graphs and random lifts

We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland’s Lower Matching Conjecture and Schrijver’s theorem. It extends the recent work of Csikvári done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of 2-li...

Counting loopy graphs with given degrees

Let d = (d1, d2, . . . , dn) be a vector of nonnegative integers. We study the number of symmetric 0-1 matrices whose row sum vector equals d. While previous work has focussed on the case of zero diagonal, we allow diagonal entries to equal 1. Specifically, for D ∈ {1, 2} we define the set GD(d) of all n × n symmetric 0-1 matrices with row sums given by d, where each diagonal entry is multiplie...

Bipartite regular graphs with fixed diameter

For given nonnegative integers k and D , we consider the problem of determining n,(k, D ) , the smallest number n for which there exists a k-regular bipartite graph on n vertices with diameter D . We solve the problem for all pairs ( k , D ) with D + 2 (mod 4) and D + 3 (mod 4), for all pairs ( k , D ) with k even or k prime and D + 3 (mod 4), for all pairs with D I 9 or k I 4, and for a few ot...

Enumerating and Counting Cycles in Bipartite Graphs