I don't know what this part of graph theory is called but I noticed in college that this is taught in the graph theory class from the mathematics department but not in the graph theory class from the computer science department.
As for adjacency matrices -- consider the (oriented) vertex-edge incidence matrix D (vertices x edges, each edge is a column, head is +1 and tail is -1, other entries are zero). Then D and D' (transpose) both have interpretations (stop and think about them; they're like gradient and divergence), as does L = D D' (graph Laplacian).
This also generalizes from graphs to simplicial complexes (triangles, tetrahedra). And "connected components" of graphs generalize to higher homology groups (loops, voids):
If the graph is an undirected graph the adjacency matrix is symmetric. People study the eigenvalue spectrum; the largest eigenvalue is bounded by the highest degree of a vertex in the graph.