Defesa de Tese.

Banca examinadora: Carlos Gustavo Moreira - Orientador - IMPA Dimitar Dimitrov - Coorientador - UNESP Rob Morris - IMPA Nicolau Saldanha - PUC-Rio Gabriel Coutinho - UFMG Eduardo Tengan - UFSC

Abstract: This thesis studies a connection between matching polynomials and continued fractions. For the matching polynomials: we prove a refinement of a theorem by Ku and Wong, which extends the classical Gallai-Edmonds decomposition; we present a generalization of Sturm's classical theorem about the number of zeros of a real polynomial in an interval; we characterize the number of distinct zeros in terms of the dimension of a vector space generated by a family of matching polynomials; we prove an upper bound on the number of paths that start at some vertex of a graph. We also present simple formulas for means of continued fractions that are useful in the classical theory of the Pell equation .