scholarly journals A Small Trivalent Graph of Girth 14

10.37236/1664 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Trivalent Graph

We construct a graph of order 384, the smallest known trivalent graph of girth 14.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

The Homology of Abelian Covers of Knotted Graphs

1999 ◽  
Vol 51 (5) ◽  
pp. 1035-1072
Author(s):  
R. A. Litherland
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Petersen Graph ◽  
Infinite Class ◽  
Colored Graphs ◽  
Branched Cover ◽  
Trivalent Graph ◽  
Abelian Covers ◽  
Index 2

AbstractLet be a regular branched cover of a homology 3-sphere M with deck group and branch set a trivalent graph Γ; such a cover is determined by a coloring of the edges of Γ with elements of G. For each index-2 subgroup H of G, MH = /H is a double branched cover of M. Sakuma has proved that H1() is isomorphic, modulo 2-torsion, to ⊕HH1(MH), and has shown that H1() is determined up to isomorphism by ⊕HH1(MH) in certain cases; specifically, when d = 2 and the coloring is such that the branch set of each cover MH → M is connected, and when d = 3 and Γ is the complete graph K4. We prove this for a larger class of coverings: when d = 2, for any coloring of a connected graph; when d = 3 or 4, for an infinite class of colored graphs; and when d = 5, for a single coloring of the Petersen graph.

scholarly journals A trivalent graph with 58 vertices and girth 9

1980 ◽  
Vol 30 (3) ◽  
pp. 299-301 ◽  
Author(s):  
N.L. Biggs ◽  
M.J. Hoare
Keyword(s):  
Trivalent Graph

scholarly journals Toric Geometry of Spin(7)-Manifolds

2019 ◽  
Author(s):  
Thomas Bruun Madsen ◽  
Andrew Swann
Keyword(s):  
Symmetry Group ◽  
Orbit Space ◽  
Moment Map ◽  
Toric Geometry ◽  
Hamiltonian Action ◽  
Trivalent Graph ◽  
Dense Set

Abstract We study $ \operatorname{Spin}(7) $-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons–Hawking type ansatz that describes such geometries in terms of a symmetric $ 4\times 4 $-matrix of functions. This description leads to the 1st known $ \operatorname{Spin}(7) $-manifolds with a rank $ 4 $ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $ \mathbb{R}^4 $ as the image of the set of the special orbits.

scholarly journals Bott–Taubes–Vassiliev cohomology classes by cut-and-paste topology

2019 ◽  
Vol 30 (10) ◽  
pp. 1950047
Author(s):  
Robin Koytcheff
Keyword(s):  
Finite Type ◽  
Integer Lattice ◽  
Configuration Spaces ◽  
Euclidean Spaces ◽  
The Real ◽  
Knot Invariants ◽  
Trivalent Graph ◽  
Higher Dimensional ◽  
Valued Graph ◽  
Knots And Links

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct “Vassiliev classes” in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in [Formula: see text] for [Formula: see text]. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-[Formula: see text] classes out of mod-[Formula: see text] graph cocycles, which need not be reductions of classes over the integers.

scholarly journals A trivalent graph of girth ten

1972 ◽  
Vol 12 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A.T Balaban
Keyword(s):  
Trivalent Graph

scholarly journals Oriented Tait graphs

1973 ◽  
Vol 16 (3) ◽  
pp. 328-331 ◽  
Author(s):  
G. Szekeres
Keyword(s):  
Planar Graph ◽  
Common Vertex ◽  
Edge Disjoint ◽  
Trivalent Graph ◽  
Edge Colouring

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.

scholarly journals Polyhedral decompositions of cubic graphs

1973 ◽  
Vol 8 (3) ◽  
pp. 367-387 ◽  
Author(s):  
G. Szekeres
Keyword(s):  
Cubic Graphs ◽  
Simple Circuit ◽  
Equivalent Formulation ◽  
Trivalent Graph

A polyhedral decomposition of a finite trivalent graphGis defined as a set of circuits= {C1, C2, …Cm} with the property that every edge ofGoccurs exactly twice as an edge of someCk. The decomposition is called even if everyCkis a simple circuit of even length. IfGhas a Tait colouring by three coloursa, b, cthen the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: ifGhas an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grünbaum) in terms of polyhedral decompositions alone.

scholarly journals New “Graphiton” Model: a Computational Discrete Space, Self-Encoded as a Trivalent Graph

2011 ◽  
Vol 5 (1) ◽  
Author(s):  
Raymond Aschheim ◽  
Smain Femmam ◽  
M. Faouzi Zerarka
Keyword(s):  
Discrete Space ◽  
Trivalent Graph
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