Geodesic Cycle Research Articles

Geodesic pancyclicity of twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. An n-dimensional twisted cube, TQ n , is an important variation of the hypercube Q n and preserves many of its desirable properties. The problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years. The geodesic cycle embedding problem is for any two distinct vertices, to find all the possible lengths of cycles that include a shortest path joining them. In this paper, we prove that TQ n is geodesic 2-pancyclic for each odd integer n ⩾ 3. This result implies that TQ n is edge-pancyclic for each odd integer n ⩾ 3. Moreover, TQ n × K 2 is also demonstrated to be geodesic 4-pancyclic.

On the cohomology of uniform arithmetically defined subgroups in SU*(2n)

AbstractWe study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

Geometric cycles, Albert algebras and related cohomology classes for arithmetic groups

We discuss the construction of totally geodesic cycles in locally symmetric spaces attached to arithmetic subgroups in algebraic groups G of type F_4 which originate with reductive subgroups of the group G . In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group is related to the automorphic spectrum of the group, this geometric construction of non-vanishing classes leads to results concerning the existence of specific automorphic forms.

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance dG(u, v) is at least dC(u, v)−e(n). Let ω(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = logd−1logd−1n+ ω(n) and |C| = 2logd−1n+ O(ω(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115-136, 2011

Geometric cycles, arithmetic groups and their cohomology

It is the aim of this article to give a reasonably detailed account of a specific bundle of geometric investigations and results pertaining to arithmetic groups, the geometry of the corresponding locally symmetric space X / Γ X/\Gamma attached to a given arithmetic subgroup Γ ⊂ G \Gamma \subset G of a reductive algebraic group G G and its cohomology groups H ∗ ( X / Γ , C ) H^{\ast }(X/\Gamma , \mathbb C) . We focus on constructing totally geodesic cycles in X / Γ X/\Gamma which originate with reductive subgroups H ⊂ G H \subset G . In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group Γ \Gamma is strongly related to the automorphic spectrum of Γ \Gamma , this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms.

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u , v ∈ V ( G ) , a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if d C ( u , v ) = max { d C ( x , y ) | x , y ∈ V ( C ) } . A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271–278] if it contains a cycle of every length from 3 to | V ( G ) | inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181–187] (respectively, balanced pancyclic) if for each pair of vertices u , v ∈ V ( G ) , it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max { 2 d G ( u , v ) , 3 } ⩽ l ⩽ | V ( G ) | . Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQ n , is pancyclic in the sense that a cycle of length l exists, 3 ⩽ l ⩽ | V ( AQ n ) | . In this paper, we study two new pancyclic properties and show that AQ n is geodesic pancyclic and balanced pancyclic for n ⩾ 2 .

On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains

Let Ω be a bounded symmetric domain equipped with a canonical Kahler metric. We are interested to characterize holomorphic geodesic cycles (i.e., compact complex geodesic submanifolds) S ⊂ X on Hermitian locally symmetric manifolds X uniformized by Ω, i.e., X = Ω/Γ, where Γ ⊂ Aut(Ω) is any torsion-free discrete group of automorphisms, in terms of differentialgeometric or algebro-geometric conditions. In Eyssidieux-Mok [EysMok1995] we studied almost geodesic complex submanifolds and formulated the gap phenomenon. Up to equivalence under Aut(Ω) there are only a finite number of totally geodesic complex submanifolds D ⊂ Ω which are themselves biholomorpic to bounded symmetric domains. Let 2 > 0. We say that S ⊂ X is 2-geodesic if and only if the norm of the second fundamental form of S ⊂ X is uniformly bounded by 2. Fixing Ω but letting Γ ⊂ Ω be arbitrary we showed that when 2 is sufficiently small, an 2-geodesic compact complex

Asymptotique de la norme $L^2$ d'un cycle géodésique dans les revêtements de congruence d'une variété hyperbolique arithmétique

Let M be an arithmetic hyperbolic manifold and \(F\subset M\) be a codimension 1 geodesic cycle. In this paper, we study the asymptotic growth of the \(L^2\)-norm of the lifts of F in the congruence tower above M. We obtain an explicit value for the growth rate of this norm. In particular, we provide a new proof of a celebrated result of Millson [Mi] on the homology of the arithmetic hyperbolic manifolds. The method is quite general and gives a new way of getting non zero homology classes in certain locally symmetric spaces.

Cycles géodésiques transverses dans les variétés hyperboliques

Let M be a hyperbolic manifold, with \( \pi_1M \) finitely generated. Let c1 and c2 be two transverse geodesic cycles with dim(c1) + dim(c2) = dim M and \( c_1 \cap c_2 \neq \emptyset \). In this paper, following ideas of [MR], we prove (Theorem 6) that c1 and c2 lifts to a finite cover of M as two submanifolds F1 and F2 with¶¶\( [F_1][F_2] \neq 0 \).¶ This theorem implies in particular that the compact hyperbolic manifolds constructed by Gromov and Piatetski-Shapiro in [GP] have nontrivial virtual Betti numbers.

Characterization of certain holomorphic geodesic cycles on quotients of bounded symmetric domains in terms of tangent subspaces

Let Ω be an irreducible bounded symmetric domain and are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.

Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables

Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincare dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero.

Locally geodesic cycles in 2-self-centered graphs

A cycle C in G is said to be locally geodesic at a vertex ν if for each vertex u on C, the distance between ν and u in C coincides with that in G. It will be shown that a self-centered graph of radius 2 contains a cycle of length 4 or 5 which is locally geodesic at each vertex and conversely that if the longest one among such cycles for each vertex of a block has length 4 then the block is self-centered and has radius 2.

Theta functions defined by geodesic cycles in quotients ofSU(p, 1)

Theta functions defined by geodesic cycles in quotients ofSU(p, 1)

Harmonic forms dual to geodesic cycles in quotients of SU (p, 1)

Harmonic forms dual to geodesic cycles in quotients of SU (p, 1)