On the two largest eigenvalues of trees

AbstractVery little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number. Starting from a 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 maximize the second largest eigenvalue. This corrects a result from the literature, and it includes a proof of a conjecture of Neumaier. The main tool for this endeavor is the theory of partial engenvectors