1. (0,1)-Matrices If the row-sums of a matrix A are r 1, …,r k, then we shall call the vector r:=(r 1,r 2, …,r k ) the row-sum of A, and similarly for the column-sums. Problem: study the existence of a (0,1)-matirx with given row-sum r and column-sum s. For convenience, assume that the coordinates of r and s are increasing. Def: Given 2 partitions r=(r 1,r 2, …,r k ) and s=(s 1,s 2, …,s m ) of the same integer N, we say that r majorizes s when r 1 +r 2 + … +r k ≥ s 1 +s 2 +…+s k for all k. Def: The conjugate of a partition r is the partition r* where r i * is the number of j such that r j ≥ i.

2. Thm 1. Let r 1, …,r n and s 1, …,s m be 2 nonincreasing sequences of nonnegative integers each summing to a common value N. There exists an n  m (0,1)-matrix with row-sum r and column-sum s iff r* majorize s. Pf: “ ⇒ ” Suppose such a matrix exists with row-sum r and column-sum s. Consider the first k columns. The number of 1’s in these columns is Thus, we have r* majorizes s. k

3. “ ⇐ ” Consider the following transportation network: Claim that a (0,1)-matrix M=(a ij ) with row-sum r and column-sum s exists iff this network admits a flow of strength N. (pf of claim:) Given such a matrix, we get a flow of strength N by saturating the edges incident with S and T and assigning flow a ij form x i to y j. By Thm 7.2, a ij can be 0 or 1. ST x1x1 x2x2 xnxn y1y1 y2y2 ymym r1r1 r2r2 rnrn s1s1 s2s2 smsm ……. 1 1 For i ∈ [n], j ∈ [m] an edge of capacity 1 from x i to y j.

4. Consider a cut (A, B) The number of edges crossing from A to B includes n-n 0 edges leaving S, m 0 edges into T and n 0 (m-m 0 ) edges from X to Y. The capacity of this cut is at least r n 0 +1 + … +r n +s m-m 0 +1 +s m-m 0 +2 + … +s m +n 0 (m-m 0 ). ……. x1x1 xn0xn0 ym0ym0 y1y1 A B S T ……

5. The number of cells in the above Ferrers diagram (of r) is N. N ≤ n 0 (m-m 0 ) + (the number of cells in the last n-n 0 row) + (the number of cells in the last m 0 columns). The number of cells in the last m 0 columns is the sum of the last m 0 parts of the conjugate r*. By the assumption that r* majorizes s, r* m-m 0 +1 + r* m-m 0 +2 + … +r* m ≤ s m-m 0 +1 + … +s m. m 0 m n 0 n

6. Thus, N ≤ n 0 (m-m 0 ) + (r n 0 +1 + … +r n ) + (r* m-m 0 +1 + … +r* m ) ≤ n 0 (m-m 0 ) + (r n 0 +1 + … +r n ) +(s m-m 0 +1 + … +s m ) ≤ (the capacity of any cut). Thus, a maximum flow achieves existence of the (0,1)- matrix. ▨

7. Another construction of (0,1)-matrix with r* majorizes s: Start with the matrix A 0 corresponding to the Ferrers diagram of r, adding columns of 0 ’ s if necessary. The column-sum of A 0 is r*. If r* majorizes s, we can find a sequence of partitions r*=s 0, s 1, …, s l =s so that each partition in the sequence is obtained from the preceding one by the operation of Problem 16A. Eg. (5, 4, 1) majorizes (3, 3, 3, 1). (5, 4, 1, 0) → (5, 3, 2, 0) → (5, 3, 1, 1) → (4, 3, 2, 1) → (3, 3, 3, 1) or (5, 4, 1, 0) → (4, 4, 1, 1) → (4, 3, 2, 1) → (3, 3, 3, 1) What is the shortest possible sequence?

8. Eg. Let r = (3, 2, 2, 2, 1) and s = (3, 3, 3, 1). Then r*=(5, 4, 1).

9. Thm 2. Given partitions r and s of an integer N, let M(r,s) denote the number of (0,1)-matrices A with row-sum r and column-sum s. If r majorizes r 0 and s majorizes s 0, then M(r 0, s 0 ) ≥ M(r, s). Pf: Fix a row-sum vector r = (r 1, …, r n ) of length n and a column-sum vector s = (s1, …, sm) of length m that have equal sums. If s1 > s2, claim M(r, (s 1 -1, s 2 +1, s 3, …, s m )) ≥ M(r, (s 1, s 2,…, s m )). (This holds for any 2 columns.) The same idea applies for fixing s. By the claim and Problem 16A, the theorem follows.

10. To prove the claim, consider (0,1)-matrices A of size nx(m-2) with column-sum (s 3, s 4, …, s m ). For a given matrix A, it may or may not be possible to append 2 columns to A to get row-sum r. If possible, we need to add two 1 ’ s to a rows one 1 to b rows, and no 1 to c rows. 1 · · · 1 1 0 0 · · · 1 · · · 1 0 1 0 · · · A a b c s 1 > s 2 a + b + c = n, 2a + b = s 1 + s 2

11. The number of ways to append the 2 columns to have column-sums s 1 and s 2 is There are to get new column-sums s 1 -1, s 2 +1. ▨

12. Cor: Let A(n,k) denote the number of nxn (0,1)-matrices with all line-sums equal to k. Then Pf: The number of (0,1)-matrices with row-sum (k, …, k) is. Each has a column-sum s = (s 1, s 2, …, s n ), 0 ≤ s i ≤ n, s 1 +s 2 + … +s n = nk, which has at most C(nk+n-1, n-1) such column-sums. Since the greatest number of associated (0,1)-matrices occurs for column-sum (k, k, …, k) ; the number of such matrices is at least the average number. Thus, ▨

13. Thm 3. Let d and d ’ be 2 partitions of an (even) integer N. If d majorizes d ’, then there are at least as many labeled simple graphs with degree sequence d ’ as with degree sequence d. Thm 4. Thm 5. Thm 6.