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ABSTRACT. We prove a Harnack inequality for eigenfunctions of certain homogeneous graphs and subgraphs which we call strongly convex. This inequality can be used to derive a lower bound for the (nontrivial) Neumann eigenvalues by l/(8kD 2 ) where k is the maximum degree and D denotes the diameter of the graph.

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Rational curves, occurring in the exceptional loci of our birational transformations, are studied in this section to prepare for what follows. The results are, however, of independent interest.A rational curve C in a projective variety W is called incompressible if it can not be deformed to a reducible or non-reduced curve.Lemma 2.1. ([2]) Let C be an incompressible rational curve in a projective algebraic variety W of dimension n. Then C moves in a family of dimension at most 2n − 2.Proof. Replacing W by its normalization if necessary, we may assume that W is normal. Then the lemma is equivalent to (1.4.4) of [2]. (One needs to observe that up to (1.4.4), the smooth assumption on the ambient variety is not used.)If there is a morphism f : P n → W such that it is an isomorphism or a