Cycles and Cuts

Abstract

In the first section of this chapter we give a rigorous derivation of the Perron–Frobenius Theorem, restricting our attention to the adjacency matrix of a graph. Some bounds for the Perron root of the adjacency matrix are obtained. As an application, we derive Turan’s Theorem on triangle-free graphs. The next section is devoted to a basic introduction to the adjacency algebra, which is the algebra generated by the adjacency matrix and its powers. For a regular graph, the adjacency matrix and the Laplacian differ only by a scalar matrix. This enables us to explore the relationship between the adjacency matrix of a regular graph and that of its complement and line graph. Several results in this direction are proved in the next section. In the final section we derive spectral properties of strongly regular graph and apply them to derive the well-known Friendship Theorem.