For most feasible spectra of connected regular graphs with four distinct eigenvalues and at most 30 vertices we find all such graphs, using both theoretic and computer results.

AMS classifications: 05E30; 05B20

AMS classification: 05C50;

We study nonregular graphs with three eigenvalues.We determine all the ones with least eigenvalue -2, and give new infinite families of examples.

The algorithms of inverse iteration and Rayleigh quotient iteration for approximating an eigenpair of a matrix contain a step in which a matrix-vector equation must be solved.The behaviour of these algorithms is analysed if this equation is solved only approximately with a known tolerance.