If you're thinking of weighted, directed graphs, a binary relation doesn't quite fully represent it. In the weighted directed case, a graph is a function from
Nodes ×
Nodes →
Weights, (where
Weights ⊆ ℤ or
Weights ⊆ ℝ or something like that)...
Which makes weighted graphs perfect for representation by matrices!
Studying adjacency matrix of weighted, directed graphs gave me a profound realization matrices are a table of relationships between their "dimensions". I never looked at matrices the same way again. I realized that the identity column was literally the entries that represented a relationship between a dimension/node and itself. I went down many rabbit holes... particularly the question of graph isomorphism. That problem is a rabbit hole. Another rabbit hole is the Hamiltonian cycle problem.
P.S. check out hypergraphs, which are not necessarily binary relations and generalize graphs. Incidentally, hypergraphs have an adjacency tensor. I've started to understand that tensors are like 'tables' of relationships between dimensions + dimensions representing some or all of their possible combinations.
By the way, if you can figure out a way to do fast multiplication of matrices over degree-truncated polynomial rings, I'll show you a fast way to count the number of hamiltonian cycles in a graph. These problems are intricately linked.