On the two largest eigenvalues of trees (Q1361770)

Very little is known about upper bounds for the largest eigenvalue of a tree that depend only on the number of vertices. Starting from the classical upper bound for the largest eigenvalue, some refinements can be obtained by successively removing trees from consideration. The results can be used to characterize those trees that maximise the second largest eigenvalue. This corrects a result in the literature, and it includes a proof of a conjecture of Neumaier. The main tool for this endeavour is the theory of partial eigenvectors.