CS 408 The Geometry of Graphs Abhiram

We usually think of a graph visually, i.e. we see vertices as points on a plane or in space, with edges as lines connecting them. When speci ed algebraically (e.g. by its adjacency matrix), the geometrical imagery is lost. Can we nd a natural embedding of a graph in the plane or in space given the adjacency matrix? This is the question for the lecture. It will turn out that such an embedding can indeed be derived. Whether it ts our bill of \naturalness" is really a subjective, almost artistic question, and unlikely to have a crisp unique answer. We will use the resulting embeddings for two purposes: drawing a graph on paper, and partitioning a graph into large parts by removing few edges. In both cases, it will be seen that our approach is reasonably satisfactory. The starting point for our construction is the edge-node incidence matrix B of a graph G with n vertices and m edges. B has n rows and m columns, and bie = 1 if vertex i is an endpoint of edge e, and 0 otherwise. Clearly, every column of B has exactly two 1s and rest 0s. Note that BB = A+D where A is the adjacency matrix, and D a diagonal matrix in which the iith entry equals the degree di of vertex i. We will later consider the oriented incidence matrix in which one of the entries in each column is negated, corresponding to an orientation of the edge. The rst point to note is that the elements of the ith row Bi of the incidence matrix B can be thought of as the coordinates of vertex i in m dimensional space. In some sense, this is a nice embedding: Bi Bj > 0 if (i; j) is an edge, and 0 otherwise. If the graph is regular, then jjBijj = jjBjjj and hence in this embedding connected vertices are seen to be closer to each other than non connected vertices. This clearly seems desirable. For non-regular graphs, the situation can be xed as follows: we de ne a normalized incidence matrix C where cie = bie= p di. Alternately C = D B. The matrix C can be seen to have the property that unconnected vertices are located farther than connected vertices. Note however, that lengths of di erent edges is di erent, whereas it was the same for the regular case. The matrices B or C place the vertices as a point cloud in m dimensional space. We will project the point cloud onto two dimensions to get a good picture of the graph, and onto a line to determine how to partition it. The key question is what direction to use for the projection.