# $M \times N$ binary matrix where $\sum ($any row$)=R$ and $\sum ($any col$)=C$

I have encountered a fairly structured $M\times N$ matrix $A$:

• $A$ has only $1$s and $0$s
• The sum along any row of $A$ equals $R$
• The sum along any column of $A$ equals $C$.
• $\frac{M}{R}=\frac{N}{C}=K$, and $K$ is an integer.

Is there a name for the matrices that satisfy these properties? What does the set of these matrices look like?

Additional info, although possibly unhelpful:

The matrix is actually a scaled discrete joint-probability distribution for finite RVs $(X,Y)$. I am interested in forming a $K$-partition of the $Y$-dimension of this matrix. Given we know what set of the partition $Y$ is in, we know the range of rows $X$ must be in. I'm trying to find the $K$-partition that minimizes the average amount of rows.

For example, one such matrix is a block-diagonal matrix with $K$ blocks of size $R\times C$, each block full of 1s. We partition the columns so that each set encapsulates the columns associated with a block. Then if we know that $Y$ is in the $i$th set, then $X$ has to be in the rows that the $i$'th block occupies. So then we know $X$ to within $R$ values.

• This is an incidence matrix, however, you still have to specify the conditions that there are $A$ ones in each row and $B$ ones in each column. – Element118 Oct 27 '15 at 7:54
• It is also the matrix that describes a bipartite graph where each vertex from one group has $R$ connections, and each vertex from the other has $C$ connections, along with that condition that involves $K$ which specifies the size of the graph. – enthdegree Oct 28 '15 at 10:13

## 1 Answer

This matrix represents a $(R,C)$-regular bipartite graph of dimension $(KR,KC)$, letting each row and column represent a vertex, and letting a '1' denote a connection between the row/column vertex.

It is also one of the set of solutions to a specific Nonogram puzzle.

Thought process continued in: Clustering large biregular graphs