Eigenvectors of Graphs
Annual rept. 1 Jul 1985-60 Jun 1986
CLARKSON UNIV POTSDAM NY
Pagination or Media Count:
Let z be an eigenvector of the adjacency matrix A of a connected graph G. Say a vertix is positive, nonnegative, zero, etc. if the same is true of the corresponding element of z. If z is an eigenvector for the second largest eigenvalue of A, it is known that the nonnegative vertices of G form a connected subgraph. This separation of vertices according to sign provides the basis for studying the structure of G as revealed by its eigenvectors, inequalities on the number of edges joining positive and negative vertices, bounds on the number of zero vertices, bounds on multiplicities and some description of the variability of the elements of z. The rows of an eigenmatrix provide a mapping of the vertices of G into m-dimensional euclidean space. Some graphs thus draw themselves. This phenomenon is especially interesting if the graph is the skeleton of a polytope.
- Theoretical Mathematics