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James Mahoney 3-2-15 Spanning Trees  Papa Bear: Too Many Edges (Contains Cycles)

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James Mahoney 3-2-15 Spanning Trees  Papa Bear: Too Many Edges (Contains Cycles) James Mahoney 3-2-15 Spanning Trees Papa Bear: Too many edges (contains cycles). Mama Bear: Too few edges (some vertices not touched). Spanning Tree: Just the right number of edges. If a graph has 푛 vertices then a spanning tree will have 푛 − 1 edges. Tree Graphs Let 퐺 be a connected graph. The tree graph of 퐺, 푇(퐺), has vertices which are the spanning trees of 퐺, where two vertices are adjacent iff you can change from one to the other by moving exactly one edge. Example: 퐶4 Example: 퐶4 Example: 퐶4 푇 퐶4 = 퐾4 Example: 퐾4 − 푒 Example: 퐾4 − 푒 Initial Investigation A graph 퐺 is hamiltonian if there is a cycle that contains every vertex of 퐺. For every graph 퐺, is 푇(퐺) hamiltonian? Can we move through all of the spanning trees of a graph just by switching one edge at a time? Known Results 푇(퐺) is hamiltonian for any graph 퐺 (Cummins, 1966) 푇 퐺 is uniformly hamiltonian (Harary & Holtzmann, 1972) 푇(퐺) is edge-pancyclic and path-full (Alspach & Liu, 1989) 휅 푇 퐺 = 휅′ 푇 퐺 = 훿(푇 퐺 ) (Liu, 1992) 휒 푇 퐺 ≤ |퐸 퐺 | (Estivill-Castro, Noy, & Urrutia, 2000) Matroid Basis Graphs Matroid Basis Graphs Tree Graphs Matroid Basis Graphs A common neighbor subgraph is the graph induced by two vertices distance two from each other and all of their common neighbors. Thm: In a MBG, every common neighbor is either a square, tetrahedron, or octahedron. (Mauer, 1973) Thm: Octahedra can’t show up in a tree graph, so every common neighbor graph of a tree graph is either a square or a tetrahedron. Graphs with Cut Vertices Let 퐺 and 퐻 be graphs and let 퐺 ⊙ 퐻 be a graph that identifies a vertex in 퐺 with a vertex in 퐻. Nearem: 푇 퐺 ⊙ 퐻 ≅ 푇(퐺)□푇(퐻). Symmetry of Tree Graphs An automorphism of a graph 퐺 is a permutation of the vertices that respects adjacency. The set of all automorphisms of 퐺 forms a group under composition, 퐴푢푡(퐺). 퐴푢푡(퐺) helps describe the symmetries and structure of 퐺. The glory of a graph 퐺, (퐺), is the size of its automorphism group. 퐺 = |퐴푢푡 퐺 |. If 퐺 is large, 퐺 is highly symmetric (glorious). 퐴푢푡(푇 퐺 ) (퐺) has been large for most of the small graphs studied so far. Thm: If 퐺 is 3-connected, then 퐴푢푡 퐺 ≅ 퐴푢푡(푇 퐺 ). Conj: The converse is true as well. Conj: If 퐺 is 2-connected but not 3-connected, (푇(퐺)) is either 1, 2, 6, or is divisible by 4. Nearem: 퐴푢푡(퐺) is a subgroup of 퐴푢푡(푇 퐺 ). 퐴푢푡 퐾4 − 푒 ≅ 푉4 while 퐴푢푡 푇 퐾4 − 푒 ≅ 퐷8, the symmetries of the square. Conj: (푇 퐺 )/(퐺) is 1, 3, or even. Automorphism Size Examples Graph 퐺 g 푻 푮 품 푮 Notes 퐾4 − 푒 8 4 퐷8 and 푉4 퐾3,2 48 12 푆4 × 푆2 and 푆3 × 푆2 퐾5 120 120 Probably 푆5 and 푆5 퐶6 ⊝ 퐶6 28800 4 ? and 푉4 288 3 ? and ℤ3 12 1 퐷12 and trivial 퐶4 24 8 푆4 and 퐷8 Isomorphic Tree Graphs Is it ever the case that 퐺 ≇ 퐻 but 푇 퐺 ≅ 푇(퐻)? Thm: If 퐺 is 3-connected and planar, 푇 퐺 ≅ 푇(퐺∗). Planar duals give isomorphic tree graphs. Halin graphs and polyhedral graphs fit this. 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. Realizing Tree Graphs Given 푇(퐺), can we find a graph 퐻 such that 푇 퐻 ≅ 푇(퐺)? Given the number of vertices and edges in a tree graph, can we put a useful bound on the number of vertices or edges in the original graph? Realizing Tree Graphs These two graphs are isoparic and their tree graphs are isoparic (both have 64 vertices and 368 edges). Deletion/Contraction Let 푒 be an edge of 퐺. Nearem: 푇 퐺 ≅ 푇 퐺 − 푒 ∪ 푇 퐺 ⋅ 푒 ∪ additional edges. Deletion/Contraction 퐺 푒 퐺 − 푒 퐺 ⋅ 푒 Planarity Diamond Butterfly Thm: If the both the diamond and the butterfly are forbidden minors, the family of graphs obtained are the pseudoforests. Pseudoforest: every connected component is a unicycle. Unicycle: a connected graph with exactly one cycle. Thm: The tree graphs of the diamond and the butterfly are nonplanar. (Contain 퐾5 and 퐾3,3 minors, respectively.) Nearem: 푇(퐺) is nonplanar unless 퐺 ≅ 퐶3, 퐶4. Pseudoforest made of unicycles Unicycles and Decomposition Each spanning tree has 푛 − 1 edges. 퐺 has 푚 edges. To find the neighbors of a vertex in 푇(퐺), add one of the 푚 − 푛 − 1 = 푚 − 푛 + 1 “extra” edges to the tree. Exactly one cycle will be formed. This unicycle of size 푐 will give rise to a 퐾푐 subgraph in 푇(퐺). Nearem: The edges of 푇(퐺) can be decomposed into cliques of size at least three such that each vertex is in exactly 푚 − 푛 + 1 cliques. Can be used to predict number of edges in 푇(퐺). Is this decomposition unique? Unicycles and Decomposition 푚 = 5 푛 = 4 푚 − 푛 + 1 = 2 Degree Bounds Let 푣 = 푚 − 푛 + 1. Thm: 푟푡ℎ 퐺 − 1 푣 ≤ 훿 푇 퐺 ≤ Δ(푇 퐺 ) ≤ 푐푟푐 퐺 − 1 푣 (Liu, 2002) By Vizing’s Theorem we then have 푟푡ℎ 퐺 − 1 푣 ≤ 휒′ 푇 퐺 ≤ 푐푟푐 퐺 − 1 푣 + 1 Additional Families Let 휃푙,푚,푛 be the graph with two vertices joined by three disjoint paths of edge length 푙, 푚, and 푛. Thm: 푇 휃푙,푚,푛 ≅ 퐿 퐾푙,푚,푛 . Nearem: This is the only time a tree graph will be a line graph. 휃4,2,1 Additional Families 푃3,4 Let 푃푛,푘 be the graph where two vertices are joined by 푛 disjoint paths of edge length 푘. Thm: 푇(푃푛,푘) is (푛 − 1)(2푘 − 1)-regular. Any number that is not a power of 2 can be written in this form. Conj: 푇(푃푛,푘) is integral (with easily-understood eigenvalues) and vertex transitive. 푇(푃푛,푘) could be a new infinite family (with two parameters) of regular integral graphs. Additional Families Bracelets: Take a graph and join together copies of it like a bracelet. Conj: If the repeated graph has a regular tree graph, the tree graph of the bracelet will be regular. Additional Families So far cycles, 푃푛,푘, and bracelets have been the only families that induce regular tree graphs. While 푇(푃푛,푘) seems to be integral, that doesn’t seem to hold for bracelets. The tree graphs of these families seem to be vertex- transitive. Conj: If 푇(퐺) is regular then it is vertex-transitive. (Zelinka, 1990) Characterizing Tree Graphs Conj: 퐻 ≅ 푇(퐺) iff it is a matroid basis graph with no induced octahedra and can be decomposed into cliques of size 3 or more, where each vertex is in the same number of cliques. Further Research Can we find good bounds for other graph parameters such as 휔 and 훼? Can we find better bounds for 휒 and Δ? Can tree graphs be characterized in a simple way? Does the number of cycles or cycle-types in 퐺 affect the regularity or integrality of 푇(퐺)? If so, how? Are there other families of regular or integral tree graphs? Thanks! Any questions? 푇(퐾4,2) .
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