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.
Authors:Carlos M. da Fonseca, Victor Kowalenko, and László Losonczi
-Capizzano , S. . , Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols , Numer. Linear Algebra Appl. , 25 ( 2018 ), e2137 .  Elsner , L. and Redheffer , R. M. , Remarks on band matrices , Numer. Math. , 10 ( 1967
This paper presents a numerical investigation of rectangular 2D waveguide problems. Thereby, the resulting Helmholtz equation is approximated by different finite elements techniques. Both homogeneous and heterogeneous material parameters are considered.