In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)= The edge isoperimetric peak of G is defined as be(G) =. Similarly the vertex isoperimetric peak of G is defined as bv(G)= .The problem of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees. Let i be an integer where . For each i define the connected edge isoperimetric value and the connected vertex isoperimetric value of G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences. In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs. Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied: (For mathematical equations pl see the pdf file)

Orientador: Sueli Irene Rodrigues Costa / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T21:17:14Z (GMT). No. of bitstreams: 1 Limberger_Roberto_M.pdf: 2247337 bytes, checksum: b1a97b81860de8b0d8d254e3a4f702d0 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho são apresentadas abordagens do problema isoperimétrico que podem ser utilizadas no ensino médio ou ensino universitário. Estas incluem: i) aspectos históricos, ii) deduções formais do problema (dentre as curvas de perímetro fixo, a circunferência é a que engloba a maior área) utilizando apenas geometria euclidiana ou via cálculo diferencial, iii) contextualização em problemas de otimização a serem abordados também utilizando recursos computacionais e iv) descrição detalhada de material audiovisual produzido para o ensino médio, com a participação do autor, para um projeto com suporte MEC - UNICAMP / Abstract: This dissertation presents approaches to the isoperimetric problem that can be used in high school or university education. These include: i)historical aspects, ii) formal deductions of the problem (among the curves of fixed perimeter, the circle encompasses most area) using only Euclidean geometry or calculus iii) contextualization in optimization problems to be also addressed using computational resources iv) detailed description of audiovisual material produced for the high school, with the participation of the author / Mestrado / Matematica / Mestre em Matemática