For a non-directed graph there will always be symmetry along the top left to bottom right diagonal. This diagonal will always be filled with zero's. These facts simplify coding. Coding a Graph Coding is concerned with storing the graph in an effecient manner. One might grab all the bits from the adjacency matrix and concatenate them to form a binary string.
Different graphs have different numbers.
Also, for undirected graphs, it suffices to concatenate the bits of the upper right triangle of the adjacency matrix as is done in the figure below. This is the graph number 46. The Adjacency list implementation typically uses less space than the adjacency matrix implementation. This method is very good for problems that involve traversing a graph. This is an example of sparse graph.