Locating the Eigenvalues of Trees

e give an O(n) method that computes, for any tree T and interval (α,β), how many eigenvalues of T lie within the interval. Our method is based on Sylvester’s Law of Inertia. We use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple. It follows that caterpillars having b back nodes, where , are not integral. We also show that among the regular caterpillars C(b,k) formed by adjoining k legs to each of b back nodes, all positive roots are in the interval , and C(b,k) is not integral if b>2.