This chapter is devoted to results which did not fit readily into earlier chapters. Section 8.1 is concerned with the behaviour of certain eigenvalues when a graph is modified, and with further bounds on the index of a graph. Section 8.2 deals with relations between the structure of a graph and the sign pattern of certain eigenvectors. Results from these first two sections enable us to give a general description of the connected graphs having maximal index or minimal least eigenvalue among those with a given number of vertices and edges. In Section 8.3 we discuss the reconstruction of the characteristic polynomial of a graph from the characteristic polynomials of its vertex-deleted subgraphs. In Section 8.4 we review what is known about graphs whose eigenvalues are integers.

More on graph eigenvalues

In this section we revisit two topics which have featured in previous chapters. The first topic concerns the relation between the spectrum of a graph G and the spectrum of some modification G′ of G. When the modification arises as a small structural alteration (such as the deletion or addition of an edge or vertex), the eigenvalues of G′ are generally small perturbations of those of G, and we say that G′ is a perturbation of G. In Subsection 8.1.1, we use algebraic arguments to establish some general rules which determine whether certain eigenvalues increase or decrease under particular graph perturbations.