In this article, the eigenvalues and eigenvectors of positive binomial operators are presented. The results generalize the previously obtained ones related to Bernstein operators. Illustrative examples are supplied.
Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q1(t)q2(t)2 ... qm(t)m, where each qi(t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of qj(t). We give an algorithm to construct the polynomials qi(t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) =
|λi| of G, where λ1, λ2, ..., λn are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic
polynomial has integral coefficients.
For vibrating strings with symmetric single-well densities, it is known that the ratio λ2/λ11 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.