CW: very nerdy
I realized a while ago that games are equivalent to a ternary valued {B,0,W} square matrix (9x9, 13x13, 19x19). With this, the shades of the stones can be arbitrarily assigned numerical values. I used B = 1 and W = -1, as black absorbs light and white reflects it, but it’s arbitrary.
Adjacency matrices are used in graph theory to indicate 1 if a vertex v of a graph v_xn is linked by an edge to a different vertex v_yn, where x and y indicate the directions of matrix (board) and would correspond to the A,B,C…,T and 1,2,3…,19 directions of the board.
Since the game of go is ternary, one could create an extended adjacency matrix for an unknown directed graph. The sign of the value could be used to indicate a direction of the connected edge, such as 1 equals “from x to y” and -1 equals “from y to x”. A black stone at, say, F9 could generate a connection between two vertices labeled F and 9, whose direction goes from F to 9, whereas a white stone at F9 would indicate a connection between the vertex F and the vertex 9 as well but going from 9 to F.
All this might be confusing to those unfamiliar with spectral graph theory, but it seems to map pretty cleanly onto it. I don’t think I can give a clear description of adjacency matrices here but there are many resources about them on the internet.
For an example, I created a matrix out of the game shown here Nine Dragons Playing With A Pearl at Sensei's Library
at move #180
(note, I might have made a mistake translating the pieces to numbers in a graph, as I did it “by hand”)
If people think this is interesting, I could make the corresponding graph for it. To avoid loops, one would need a graph of < 19 + 19 = 38 vertices. Without loops, it is what is known as a Directed Acyclic Graph or DAG.
Since go games evolve through time, the graph would also change with each discrete time step (move played), and would become what is known as a temporal network.
This might have been noticed before, but it was new to me. A dedicated person might be able to analyze go games in a possibly novel way by looking at their graph-dual.