Figures

Graph G and Adjacency Matrix A:

$$A(G)= \begin{array}{c} 1\\ 2\\ 3\\ 4\\ \end{array}\left[ \be... ...&1&0&0\\ 1&0&0&1\\ 0&0&0&1\\ 0&1&1&0\\ \end{array} \right]$$

Bipartite Graph G and Adjacency Matrix:

$$A(G)= \begin{array}{c} 1\\ 2\\ 3\\ 4\\ 5\\ \end{array}\left[... ...0&0&1&0\\ 0&1&0&0\\ 0&0&1&0\\ 1&1&0&1\\ \end{array}\right]$$

Tournament(K) and its Matrix A(K)

$$A(K)= \begin{array}{c} x_1\\ x_2\\ x_3\\ x_4\\ x_5\\ \end{ar... ...0\\ 0&0&0&1&1\\ 1&0&0&0&0\\ 1&1&0&1&0\\ \end{array}\right]$$

Tournament(K) and its Matrix A^T(K)

$$A^T(K)= \begin{array}{c} x_1\\ x_2\\ x_3\\ x_4\\ x_5\\ \end{... ...1\\ 1&1&0&0&0\\ 1&1&1&0&1\\ 0&0&1&0&0\\ \end{array}\right]$$

For a Tournament(K): A(K)+A^T(K)=J-I

$$\left[ \begin{array}{ccccc} 0&1&1&1&0\\ 0&0&1&1&0\\ 0&0&0&1... ...1\\ 1&1&0&1&1\\ 1&1&1&0&1\\ 1&1&1&1&0\\ \end{array}\right]$$

Boolean Rank, b(A), of a {0,1}-matrix A

$$A= \left[ \begin{array}{cccccc} 0&1&0&1&1&1\\ 0&0&0&1&1&1\\ ... ...ay}{cccccc} 0&0&0&1&1&1\\ 0&1&0&1&0&0\\ \end{array}\right]=$$

$$\left[ \begin{array}{c} 1\\ 1\\ 1\\ 0\\ 0\\ 0\\ \end{array} ... ...left[ \begin{array}{cccccc} 0&1&0&1&0&0\\ \end{array}\right]=$$

$$\left[ \begin{array}{cccccc} 0&0&0&1&1&1\\ 0&0&0&1&1&1\\ 0&... ...&0&0&0&0&0\\ 0&1&0&1&0&0\\ 0&1&0&1&0&0\\ \end{array}\right]$$

Non-negative Integer Rank, r_z⁺(A), of A(D)

$$A= \left[ \begin{array}{cccccc} 0&1&0&1&1&1\\ 0&0&0&1&1&1\\ ... ...0&0&1&1&1\\ 0&1&0&0&0&0\\ 0&0&0&1&0&0\\ \end{array}\right]=$$

$$\left[ \begin{array}{c} 1\\ 1\\ 1\\ 0\\ 0\\ 0\\ \end{array}\... ...left[ \begin{array}{cccccc} 0&0&0&1&0&0\\ \end{array}\right]=$$

$$\left[ \begin{array}{cccccc} 0&0&0&1&1&1\\ 0&0&0&1&1&1\\ 0&... ...&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&1&0&0\\ \end{array}\right]$$

Independent 1's, t(A), of a matrix A

$$A= \left[ \begin{array}{ccccc} {\bf\underline{1}}&1&0&1&0\\ ... ... 0&0&0&{\bf\underline{1}}&0\\ 0&1&0&0&0\\ \end{array}\right]$$

Isolated 1's of a matrix A

$$A=\left[\begin{array}{cccc} 0&{\bf\underline{1}}&1&1\\ {\bf\underline{1}}&0&1&1\\ 1&1&0&1\\ 1&1&1&0\\ \end{array}\right]$$