In this paper, we consider the problem: what kinds of Riemann surfaces S will have rst eigenvalue 1 (S) large? Bye large, we mean that there is some positive lower bound for S independent of the genus. There are a number of constructions of surfaces with 1 large, the main example being the quotients of the hyperbolic plane by the congruence subgroups of PSL(2; Z). Most of the known examples rel...

1.1. The independence ratio and the minimum eigenvalue. An independent set is a set of vertices in a graph, no two of which are adjacent. The independence ratio of a graph G is the size of its largest independent set divided by the total number of vertices. If G is regular, then the independence ratio is at most 1/2, and it is equal to 1/2 if and only if G is bipartite. The adjacency matrix of ...

Let G be a connected graph with least eigenvalue −2, of multiplicity k. A star complement for −2 in G is an induced subgraph H = G − X such that |X | = k and −2 is not an eigenvalue of H . In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of −2. In some instances, G itself can be characterized by a star complement. If G is not...