# Combinatorics

Let be a bipartite graph with partitions .

Suppose

where .

Then, for non-empty $A ⊂ U,

Equality holds iff is (r, s)-regular and .

Let $$$ be the subgraph induced by , then

where is the number of edges in

Let be a bipartite graph with partitions .

Suppose

where .

Then, for non-empty $A ⊂ U,

Equality holds iff is (r, s)-regular and .

Let $$$ be the subgraph induced by , then

where is the number of edges in