5-5-2016 Acknowledgements Dr. John Caughman, Chair Committee - PowerPoint PPT Presentation

James Mahoney Dissertation Adviser: Dr. John Caughman Portland State University 5-5-2016

Acknowledgements  Dr. John Caughman, Chair  Committee members:  Dr. Nirupama Bulusu  Dr. Derek Garton  Dr. Paul Latiolais  Dr. Joyce O’Halloran  PSU Math Department, Enneking Fellowship Comm.  Friends and Family

Introduction – Background – T(G) function – Properties - Matchings Overview Introduction 1. Background 2. The Tree Graph Function and Parameters 3. Properties of Tree Graphs 4. Trees and Matchings in Complete Graphs 5.

Introduction – Background – T(G) function – Properties - Matchings Introduction  Tree graphs first introduced by Cummins in 1966  ~20 major papers published since then  No one has systematically constructed them before  My two years of research builds on data from dozens of examples

Introduction – Background – T(G) function – Properties - Matchings Overview Introduction 1. Background 2. The Tree Graph Function and Parameters 3. Properties of Tree Graphs 4. Trees and Matchings in Complete Graphs 5.

Introduction – Background – T(G) function – Properties - Matchings Graphs and Spanning Trees  Graphs have vertices and edges  Trees are connected graphs with no cycles  Spanning trees have the same vertices as the original graph  If a graph has 𝑜 vertices then a spanning tree will have 𝑜 − 1 edges

Introduction – Background – T(G) function – Properties - Matchings Tree Graphs  Let 𝐻 be a graph. The tree graph of 𝐻 , 𝑈(𝐻) , has vertices which are the spanning trees of 𝐻 , where two vertices are adjacent if and only if you can change from one to the other by moving exactly one edge.

Introduction – Background – T(G) function – Properties - Matchings Example: 𝐷 4

Introduction – Background – T(G) function – Properties - Matchings Example: 𝐷 4

Introduction – Background – T(G) function – Properties - Matchings Example: 𝐷 4 𝑈 𝐷 4 = 𝐿 4

Introduction – Background – T(G) function – Properties - Matchings Overview Introduction 1. Background 2. The Tree Graph Function and Parameters 3. Properties of Tree Graphs 4. Trees and Matchings in Complete Graphs 5.

Introduction – Background – T(G) function – Properties - Matchings Tree Graph Function & Parameters  Thm (Liu, 1992): = 𝜆 ′ 𝑈 𝐻 𝜆 𝑈 𝐻 = 𝜀(𝑈 𝐻 )  Tree graphs are as connected as possible - hard to break apart by removing vertices or edges

Introduction – Background – T(G) function – Properties - Matchings Graphs with Cut Vertices  Let 𝐻 and 𝐼 be graphs and let 𝐻 ⊙ 𝐼 be a graph that joins a vertex in 𝐻 with a vertex in 𝐼 .  Thm : 𝑈 𝐻 ⊙ 𝐼 ≅ 𝑈(𝐻)□𝑈(𝐼) .  Tree graphs of joined graphs are the product of the tree graphs of the pieces

Introduction – Background – T(G) function – Properties - Matchings Realizing Tree Graphs  Given 𝑈(𝐻) , can we find a graph 𝐼 such that 𝑈 𝐼 ≅ 𝑈(𝐻) ?  What is the pre-image of a tree graph? Where do I come from?

Introduction – Background – T(G) function – Properties - Matchings Isomorphic Tree Graphs  These pairs of graphs are not isomorphic, but their tree graphs are.  The starting graphs are isoparic : they have the same number of vertices and same number of edges but are not isomorphic.

Introduction – Background – T(G) function – Properties - Matchings Isomorphic Tree Graphs  These pairs of graphs are not isomorphic, but their tree graphs are.  The starting graphs are isoparic : they have the same number of vertices and same number of edges but are not isomorphic.

Introduction – Background – T(G) function – Properties - Matchings Realizing Tree Graphs  These two graphs are isoparic and their tree graphs are isoparic (both have 64 vertices and 368 edges).

Introduction – Background – T(G) function – Properties - Matchings Isomorphic Tree Graphs  Is it ever the case that 𝐻 ≇ 𝐼 but 𝑈 𝐻 ≅ 𝑈(𝐼) ?  Thm : If 𝐻 is 3-connected and planar, 𝑈 𝐻 ≅ 𝑈(𝐻 ∗ ) . Planar duals give isomorphic tree graphs.

Introduction – Background – T(G) function – Properties - Matchings Tree Graph Function Tree Graphs Isoparic Isomorphic Neither Isoparic Starting Graphs Isomorphic Never Always Never Neither ? Default Non planar duals?

Introduction – Background – T(G) function – Properties - Matchings Overview Introduction 1. Background 2. The Tree Graph Function and Parameters 3. Properties of Tree Graphs 4. Trees and Matchings in Complete Graphs 5.

Introduction – Background – T(G) function – Properties - Matchings Properties of Tree Graphs  Thm (Cummins, 1966): 𝑈(𝐻) is hamiltonian for any graph 𝐻  There is a cycle through all of the vertices

Introduction – Background – T(G) function – Properties - Matchings Symmetry of Tree Graphs  An automorphism of a graph 𝐻 is a permutation of the vertices that respects adjacency. The set of all automorphisms of 𝐻 forms a group under composition, 𝐵𝑣𝑢(𝐻) .  The glory of a graph 𝐻 , 𝑕(𝐻) , is the size of its automorphism group. 𝑕 𝐻 = |𝐵𝑣𝑢 𝐻 | .  𝑕(𝐻) has been large for most of the small graphs studied so far.

Introduction – Background – T(G) function – Properties - Matchings 𝐵𝑣𝑢(𝑈 𝐻 )  Thm : 𝐵𝑣𝑢(𝐻) is a subgroup of 𝐵𝑣𝑢(𝑈 𝐻 ) .  The symmetries of the input are mirrored in the symmetries of the output.  Example: 𝐵𝑣𝑢 𝐿 4 − 𝑓 ≅ 𝑊 4 while 𝐵𝑣𝑢 𝑈 𝐿 4 − 𝑓 ≅ 𝐸 8 , the symmetries of the square.

Introduction – Background – T(G) function – Properties - Matchings Summary of Proof  Every graph automorphism 𝜏 of 𝐻 induces a tree graph automorphism 𝜚 𝜏 of 𝑈(𝐻)  If 𝜚 𝜏 fixes all vertices of 𝑈(𝐻) , then 𝜏 fixes all cycle edges of 𝐻  In a 2-connected graph, all edges are cycle edges  If all edges of 𝐻 are fixed by 𝜏 , all vertices are fixed also  Therefore map that takes 𝜏 to 𝜚 𝜏 is an injective homomorphism 𝐵𝑣𝑢(𝑈 𝐻 ) 𝐵𝑣𝑢(𝐻)

Introduction – Background – T(G) function – Properties - Matchings Automorphism Size Examples Graph 𝐻 𝒉 𝑯 g 𝑼 𝑯 Notes 𝐸 8 and 𝑊 8 4 4 𝑇 4 × 𝑇 2 and 𝑇 3 × 𝑇 2 𝐿 3,2 48 12 𝐿 5 𝑇 5 and 𝑇 5 120 120 ? and 𝑊 28800 4 4 ? and ℤ 3 288 3 𝐸 12 and trivial 12 1 𝐷 4 𝑇 4 and 𝐸 8 24 8

Introduction – Background – T(G) function – Properties - Matchings Planarity  Thm : The tree graphs of the diamond and the butterfly are nonplanar. (Contain 𝐿 5 and 𝐿 3,3 minors, respectively.)  Thm : 𝑈(𝐻) is nonplanar unless 𝐻 ≅ 𝐷 3 , 𝐷 4 .  Cannot draw them flat without lines crossing. Diamond Butterfly 𝑈 𝐼 ≤ 𝑈 𝐻 𝐼 ⊑ 𝐻

Introduction – Background – T(G) function – Properties - Matchings Decomposition  Thm : The edges of 𝑈(𝐻) can be decomposed into cliques of size at least three such that each vertex is in exactly 𝑛 − 𝑜 + 1 cliques.  Can break apart graph into pieces that are completely connected, where each vertex is in same number of pieces.  Can be used to predict number of edges in 𝑈(𝐻) .

Introduction – Background – T(G) function – Properties - Matchings Decomposition 𝑛 = 5 𝑜 = 4 𝑛 − 𝑜 + 1 = 2

Introduction – Background – T(G) function – Properties - Matchings Additional Families 𝑄 3,4  Let 𝑄 𝑜,𝑙 be the graph where two vertices are joined by 𝑜 disjoint paths of edge length 𝑙 .  Thm : 𝑈(𝑄 𝑜,𝑙 ) is (𝑜 − 1)(2𝑙 − 1) -regular.  Conj : 𝑈(𝑄 𝑜,𝑙 ) is integral (with easily-understood eigenvalues) and vertex transitive.  𝑈(𝑄 𝑜,𝑙 ) could be a new infinite family (with two parameters) of regular integral graphs.  These are really nice graphs 𝑈(𝑄 3,2 )

Introduction – Background – T(G) function – Properties - Matchings Overview Introduction 1. Background 2. The Tree Graph Function and Parameters 3. Properties of Tree Graphs 4. Trees and Matchings in Complete Graphs 5.

Introduction – Background – T(G) function – Properties - Matchings Def . A perfect matching is a set of disjoint edges that covers all of the vertices in a graph. Doyle Graph Coxeter Graph

Introduction – Background – T(G) function – Properties - Matchings Coloring the edges of a graph 6 1 A coloring is an assignment of colors (numbers) to the edges of a graph 5 2 A proper coloring has distinct 4 colors at each vertex. 5 Notice that the color classes for a proper 3 coloring must be disjoint sets of edges 2 (= matchings!)

Introduction – Background – T(G) function – Properties - Matchings 1-factorizations of 𝐿 2𝑜  Lots of not-so- nice ones… In fact, of the 396 different rainbow colorings of 𝐿 10 , most look ‘random’  Some very nice ones… The most commonly known rainbow coloring of 𝐿 2𝑜 is called 𝐻𝐿 2𝑜