Cohomology of GKM fiber bundles

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray–Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.     

Realizability of Graphs

A graph is d-realizable if, for every configuration of its vertices in E^N, there exists a another corresponding configuration in E^d with the same edge lengths. A graph is 2-realizable if and only if it is a partial 2-tree, i.e., a subgraph of the 2-sum of triangles in the sense of graph theory. We show that a graph is 3-realizable if and only if it does not have K₅ or the 1-skeleton of the octahedron as a minor.     

The hard Lefschetz property for Hamiltonian GKM manifolds

We introduce characteristic numbers of a graph and demonstrate that they are a combinatorial analogue of topological Betti numbers. We then use characteristic numbers and related tools to study Hamiltonian GKM manifolds whose moment maps are in general position. We study the connectivity properties of GKM graphs and give an upper bound on the second Betti number of a GKM manifold. When the manifold has dimension at most 10, we use this bound to conclude that the manifold has nondecreasing even Betti numbers up to half the dimension, which is a weak version of the Hard Lefschetz Property.     

Geometric graphs which are 1-skeletons of unstacked triangulated polygons

We give sufficient (and necessary) conditions of local character ensuring that a geometric graph is the 1-skeleton of an unstacked triangulation of a simple polygon.     

Graphs which are locally a cube

We prove that there are exactly two connected graphs which are locally a cube: a graph on 15 vertices which is the complement of the (3×5)-grid and a graph on 24 vertices which is the 1-skeleton of a certain 4-dimensional regular polytope called the 24-cell.     

Finite Free Resolutions and 1-Skeletons of Simplicial Complexes

A technique of minimal free resolutions of Stanley—Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d–1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d–1 is d-connected.     

Colorful polytopes and graphs

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P _G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P _G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.     

A GKM description of the equivariant cohomology ring of a homogeneous space

Let T be a torus of dimension n > 1 and M a compact T-manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/T is zero dimensional and the set of one dimensional orbits in M/T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph. In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.     

Cyclic crystallizations of spheres

If the colored graph of a crystallization embeds regularly in a specific pseudosurface, then it is a sphere. This gives a characterization of the crystallizations of the 3-sphere with unknotted 1-skeleton.     

Complete Classifications of Generalized Kac–Moody Algebras Possessing Special Imaginary Roots and Strictly Imaginary Property

In this article, we consider all generalized Kac–Moody algebras (GKM algebras) for the purpose of finding out special imaginary roots and strictly imaginary roots. We give a complete classification of GKM algebras possessing special imaginary roots and also give a complete classification of GKM algebras possessing strictly imaginary property (GKM algebras whose imaginary roots are strictly imaginary). We also conclude that the Monster Lie algebra has no special imaginary root and does not satisfy strictly imaginary property.     

Heptavalent Symmetric Graphs with Certain Conditions

A graph Γ is said to be symmetric if its automorphism group Aut(Γ)acts transitively on the arc set of Γ.We show that if Γ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group G of automorphisms,then either G is normal in Aut(Γ),or Aut(Γ)contains a non-abelian simple normal subgroup T such that G≤T and(G,T)is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups.If G is arc-transitive,then G is always normal in Aut(r),and if G is regular on the vertices of Γ,then the number of possible exceptional pairs(G,T)is reduced to 5.     

Non-Abelian Hopf Cohomology of Radford Products

We study the non-abelian Hopf cohomology theory of Radford products with coefficients in a comodule algebra. We show that these sets can be expressed in terms of the non-abelian Hopf cohomology theory of each factor of the Radford product. We write down an exact sequence relating these objects. This allows to compute explicitly the non-abelian Hopf cohomology sets in large classes of examples.     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

Dihedral Branches Coverings of Hyperbolic Orbifolds

We consider the class of hyperbolic 3-orbifolds whose underlying topological space is the 3-sphere S ³ and whose singular set is a trivalent graph with singular index 2 along each edge (an important special case occurs when the trivalent graph is the 1-skeleton of a hyperbolic polyhedron). Our main result is a classification of the D-branched coverings of these orbifolds (where D₂ is the dihedral group of order 4) under some general conditions on their isometry groups or the lengths of their geodesics.     

Cayley Snarks and Almost Simple Groups

A Cayley snark is a cubic Cayley graph which is not 3-edge-colourable. In the paper we discuss the problem of the existence of Cayley snarks. This problem is closely related to the problem of the existence of non-hamiltonian Cayley graphs and to the question whether every Cayley graph admits a nowhere-zero 4-flow. So far, no Cayley snarks have been found. On the other hand, we prove that the smallest example of a Cayley snark, if it exists, comes either from a non-abelian simple group or from a group which has a single non-trivial proper normal subgroup. The subgroup must have index two and must be either non-abelian simple or the direct product of two isomorphic non-abelian simple groups. Received January 18, 2000 Research partially supported by VEGA grant 1/3213/96 Research partially supported by VEGA grants 1/3213/96 and 1/4318/97     

The minimum semidefinite rank of a triangle-free graph

We employ a result of Moshe Rosenfeld to show that the minimum semidefinite rank of a triangle-free graph with no isolated vertex must be at least half the number of its vertices. We define a Rosenfeld graph to be such a graph that achieves equality in this bound, and we explore the structure of these special graphs. Their structure turns out to be intimately connected with the zero-nonzero patterns of the unitary matrices. Finally, we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank, and provide a conjecture in this direction.     

一种关于图的关联矩阵秩的定理证明的新方法

关于图的关联矩阵的一个重要的定理是r个结点的连通图G的关联矩阵的秩是r-1.利用一般域上的线性空间理论,给出了无向图的关联矩阵秩的定理证明,该方法结构严谨且利于学生理解和接受.     

On the rank spread of graphs

For a simple graph G?=?(𝒱, ?) with vertex-set 𝒱?=?{1,?…?,?n}, let 𝒮(G) be the set of all real symmetric n-by-n matrices whose graph is G. We present terminology linking established as well as new results related to the minimum rank problem, with spectral properties in graph theory. The minimum rank mr(G) of G is the smallest possible rank over all matrices in 𝒮(G). The rank spread r _v (G) of G at a vertex v, defined as mr(G)???mr(G???v), can take values ??∈?{0,?1,?2}. In general, distinct vertices in a graph may assume any of the three values. For ??=?0 or 1, there exist graphs with uniform r _v (G) (equal to the same integer at each vertex v). We show that only for ??=?0, will a single matrix A in 𝒮(G) determine when a graph has uniform rank spread. Moreover, a graph G, with vertices of rank spread zero or one only, is a λ-core graph for a λ-optimal matrix A in 𝒮(G). We also develop sufficient conditions for a vertex of rank spread zero or two and a necessary condition for a vertex of rank spread two.     

A new asymmetric inclusion region for minimum weight triangulation

As a global optimization problem, planar minimum weight triangulation problem has attracted extensive research attention. In this paper, a new asymmetric graph called one-sided β-skeleton is introduced. We show that the one-sided circle-disconnected (?2b){(\sqrt{2}\beta)} -skeleton is a subgraph of a minimum weight triangulation. An algorithm for identifying subgraph of minimum weight triangulation using the one-sided (?2b){(\sqrt{2}\beta)} -skeleton is proposed and it runs in O(n^4/3+e+min{klogn, n²logn}){O(n^{4/3+\epsilon}+\min\{\kappa \log n, n^2\log n\})} time, where κ is the number of intersected segmented between the complete graph and the greedy triangulation of the point set.