Figure 9: Graphs representing relationships between Alice and Cheshire Cat. |

\begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 0 \\ \end{array} \right. & \left. \begin{array}{c} 0\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Cheshire doesn't like himself in the graph in Figure 9 (b). But, I felt he was a bit kind to Alice in the book. Cheshire Cat might like Alice. Here I assume Alice doesn't like Cheshire Cat, but Cheshire Cat likes Alice. Such a situation is represented by the following adjacency matrix: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right. & \left. \begin{array}{c} 0\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Each element of the adjacency matrix means following: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} \mbox{A $\rightarrow$ A} \\ \mbox{C $\rightarrow$ A} \\ \end{array} \right. & \left. \begin{array}{c} \mbox{A $\rightarrow$ C}\\ \mbox{C $\rightarrow$ C}\\ \end{array} \right] \end{array} \end{eqnarray*} Where ``A'' is Alice and ``C'' is Cheshire Cat. We read ``A \(\rightarrow\) A''as ``Alice likes Alice,'' ``A \(\rightarrow\) C'' as ``Alice likes Cheshire Cat.'' Now you can make an adjacency matrix that represents any of the graphs in Figure 9. The graph of Figure 9 (c) means that Alice likes herself and Cheshire Cat, and that Cheshire Cat doesn't like himself, but that he likes Alice. The adjacency matrix of this scenario is the following: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right. & \left. \begin{array}{c} 1\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Please note here that a mutual like-relationship becomes an undirected edge (Figure 9 (d)) and the adjacency matrix has a special form --- it is called a

*symmetry matrix*. In this case, if you switch the positions of Alice and Cheshire Cat, the matrix stays the same. The elements of the matrix are symmetrical along the

*diagonal*--- the line formed by the elements from the upper left to the lower right --- as if they are reflected in a mirror.

## No comments:

Post a Comment