Why incidence and adjacency matrices (graph theory)h

[Avichal]

My book introduces the concept of adjacency and incidence matrices but I don't understand its use.
Normally we shift from mathematical symbols and representation to graphical interpretation like in Cartesian graphs - to visualize functions better we draw them on a graph.
But here we are doing the opposite. From nice graphs we are shifting towards matrices that do not help us much visually

 

[Stephen Tashi]

Normally we shift from mathematical symbols and representation to graphical interpretation like in Cartesian graphs - to visualize functions better we draw them on a graph.

That's false. For example, we generally don't compute derivative of a function by graphing it and then trying to do some geometric construction on a graph. My impression of visual presentations is that they are rather like decorations that accompany the mainstream of mathematics as it follows a mostly symbolic course.

As to graph theory, what would a visual proof about a graph with 5,000 nodes look like? What are the rules of the game for visual proofs? Do you say, "See, you can look at the picture and tell that..."?