Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator

LetE be a Banach lattice which consists of some functionsu: over a fixed domain . This article is concerned with the local stability of a non-zero positive solutionu ^*E to the Hammerstein equationu(x)= x, wherek0, andf:×[0, )[0, ) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and forx,u>0. Under some mild additional hypotheses onE, k, andf it is proved that the spectral radius of the Fréchet derivative atu ^* of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

A survey of partial difference sets

LetG be a finite group of order . Ak-element subsetD ofG is called a (,k, , )-partial difference set if the expressionsgh ^–1, forg andh inD withgh, represent each nonidentity element inD exactly times and each nonidentity element not inD exactly times. IfeD andgD iffg ^–1D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.     

Dense graphs and edge reconstruction

By a well-known result of Nash-Williams if a graphG is not edge reconstructible, then for all ,|A||E(G)| mod 2 we have a permutation ofV(G) such thatE(G)E(G)=A. Here we construct infinitely many graphsG having this curious property and more than edges.Research (partially) supported by Hungarian National Foundation for Scientific Research Grant No.T016389.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

On signpost systems and connected graphs

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P W × W × W and the following statements hold: if (u, v, w) P, then (v, u, u) P and (v, u, w) P, for all u, v, w W; if u v; then there exists r W such that (u, r, v) P, for all u, v W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z W: if (u, v, x), (u, v, z), (x, y, z) P, then (u, v, y) P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y W: if (u, v, x), (x, y, v) P, then (u, v, y), (x, y, u) P.By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v W: u and v are adjacent in G if and only if (u, v, v) P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.     

An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let denote a distance-regular graph with diameter D 3, valency k, and intersection numbers a _i, b _i, c _i. Let X denote the vertex set of and fix x X. Let denote the vertex-subgraph of induced on the set of vertices in X adjacent X. Observe has k vertices and is regular with valency a ₁. Let _{1 2} ··· _k denote the eigenvalues of and observe ₁ = a ₁. Let denote the set of distinct scalars among ₂, ₃, ..., _k . For let mult denote the number of times appears among ₂, ₃,..., _k . Let denote an indeterminate, and let p ₀, p₁, ...,p _D denote the polynomials in [] satisfying p ₀ = 1 andp _i = c _i+1 p _i+1 + (a _i – c _i+1 + c _i)p _i + b _i p _i–1 (0 i D – 1),where p _–1 = 0. We show where we abbreviate = –1 – b ₁(1+)^–1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat _X ( ) generated by A, E*₀, E*₁, ..., E* _D , where A denotes the adjacency matrix of and E* _i denotes the projection onto the ith subconstituent of with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE* _i W 1 for 0 i D. By the endpoint of W we mean min{i|E* _i W 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 i D – 1; (ii) Equality holds in the above inequality for i = D – 1; (iii) Every irreducible T-module with endpoint 1 is thin.     

Integration by parts for a Lie group valued Brownian motion

LetG be a Lie group ofd×d matrices and be theLLie algebra ofG. We choose some Euclidean norm on , and an orthonormal basis (D ₁,...D _m ) relative to it. Let be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the Laplacian .     

G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates

In Ref. 1, we considered theG-closure of some initially given arbitrary setU of the positive-definite, symmetrical plane tensorsD of the 2nd rank, connected with the differential operator ·D · in two dimensions. Here, theG-closure procedure is applied to the 4th-order operator ··D ·· in a plane, arising in the theory of plates and containing self-adjoint tensorsD of the 4th rank. The paper generalizes some results obtained earlier in Refs. 2 and 3. The complete solution of the general problem of regularization, which presupposes the arbitrary character of the initially given setU, is not yet obtained.     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

Opérateurs différentiels associés à certaines représentations unitaires d'un groupe de Lie résoluble exponentiel

Soient G = exp g un groupe de Lie résoluble exponentiel et H = exp h un sous-groupe connexe de G. Soient un caractère unitaire de H et = Ind_H ^G. Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalente à la commutativité de D (G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre D (G) ^H , des opérateurs différentiels qui laissent l'espace des vecteurs C de invariant et qui commuttent avec l'action de H sur cet espace.     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Orientation reversing involutions and the first betti number for finite coverings of 3-manifolds

LetMS ³,P ³ be a closed, orientable irreducible 3-manifold which admits an orientation reversing involution :MM. If dim(Fix )=0, suppose ₁ (M) has a subgroup of even index. We show thatM has finite coverMMM} with ₁(M<0). As an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number.     

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let be the fundamental group of a closed orientable surface of genus g 1, and let R(, G)/G be the space of conjugacy classes of representations of into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map ¯ on R(, G)/G and investigate whether the fibres of ¯ are isotropic with respect to the natural symplectic structure on R(, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of ¯ are also isotropic, but we give an example to show that in general they are not.