Presentations for (G,s)-transitive graphs of small valency
Given a group G , a subgroup H and an element a ∈ G , we define Γ( G, H, a ) to be the graph on the set of left-cosets of H in G , where two left-cosets g 1 H and g 2 H are adjacent whenever . We will consider this construction only in the case that The first of these conditions assures that adjacency is a symmetric relation (i.e. that Γ( G, H, a ) is an undirected graph) and the second assures that Γ( G, H, a ) is connected.

title: Extended generalized hexagons

title: On Distance-Transitive Graphs
abstract: On suppose que Γ est un graphe non oriente connexe de genre fini g et G est un sous-groupe de aut(Γ) tel que pour chaque sommet x, G(x) est fini et agit transitivement sur l'ensemble des s-chemins (x 0 ,…,x s ) avec x 0 =x. Soit M un entier positif avec M≡3 (mod 8) et soit m=(M+9)/4 et n=max(2m+4,14). Si s≥2(m−1) et s≥(g−M)/2 alors Gm(x 0 ,…,x n )=1 pour chaque n-chemin (x 0 ,…,x n )

title: The nonexistence of 8-transitive graphs
abstract: We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.

title: On finite Moufang polygons
abstract: We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG1(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y).

title: 3-transpositions in infinite groups
abstract: Let G be a group. A subset D will be called a set of 3-transpositions if | x | = 2 for x e D and | xy | = 3 whenever x, yeD do not commute. We will call the set D closed if xDx −1 = D for each xeD . For each xeD , let For each subset X of D , we denote by [ X ] the graph with vertex set X where two elements x, yeX are joined by an edge whenever they commute. We denote by ( X ) the complement graph; thus two elements x, yeX are joined by an edge of ( X ) whenever they do not commute.

title: Bifurcation in difference approximations to two-point boundary value problems
abstract: Numerical methods for bifurcation problems of the form (*) LyAXf(y), By =0, where f(0) = 0 and f'(0) * 0, are considered. Here y is a scalar function, X is a real scalar, L is a linear differential operator and By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if (*) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of (*) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.

title: Symmetric graphs with projective subconstituents
abstract: Let P be an undirected graph and G a subgroup of aut(r) acting transitively on the vertex set of P. Let x be an arbitrary vertex of P. We denote by P(x) the set of vertices adjacent to x and by G(x)F(x) the permutation group induced by the stabilizer G (x) of x in G on P(x); G (x)r(X) is called the subconstituent of the pair (P, G). For each i E N let Gi(x) = {a E G(x)Ia E G(y) if a (x, y) 1 then s 3. In , it was shown that s 3. We prove the following partial result:

title: On the Approximation of Fixed Points of Nonlinear Compact Operators
abstract: Certain methods for the approximate solution of $x - Kx = 0$, where K is a compact operator, lead to a family of operator equations $x - K_n x = 0$, $n \geqq 1$, where $K_n $,$n \geqq 1$, are collectively compact and converge pointwise to K. In this paper a new and simplified proof of the existence and uniqueness of a solution of $x - K_n x = 0$ in a neighborhood of an isolated solution of $x - Kx = 0$ is given.

title: A Geometric Characterization of Certain Groups of Lie Type
abstract: Let r be an undirected graph with vertex set V(r) and edge set E(r) and let G be a subgroup of Aut (r). For each {x, y} E E(r), we denote by G{x, y} the stabilizer of {x, y} in G (which mayor may not contain elements exchanging the vertices x and y). For each x E V(r) we denote by r(x) the set of vertices adjacent to x, by G(x) the stabilizer of x in G and, for each i E N, by Gi(x) the subgroup {a E Gla E G(u) for all u E V(r) with a(x, u) "'" i} where a(x, u) denotes the distance from x to u. An s-path (for sEN) is an (s + I)-tuple (xo, Xl. .•• , xs ) of vertices such that Xi E r(Xi-l) if 1"", i "'" s and Xi ¥Xi-Z if 2"", i "'" s. Let G(xo, ... , xs ) = G(xo) n··· n G(xs ) and Gi(xo, ... , xs ) = Gi(xo) n· .. n Gi(xs ) for each s-path (xo, ... , xs ) and each i E N. If H is any group acting on a set X, we denote by H X the permutation group on X induced by H. . In  we proved the following result.

title: Permutation groups with projective unitary subconstituents
abstract: AssTRAcT. Let r be a finite directed graph with vertex set V(I) and edge set EQ7) and let G be a subgroup of aut(I) which we assume to act transitively on both V(r) and E(r). Suppose that for some prime power q, the stabilizer G(x) of a vertex x induces on both {yI(x,y) E E(r)) and {wl(w, x) E E)} a group lying between PSU(3, q2) and Pr1U(3, q2). It is shown that if G acts primitively on V(I), then for each edge (x, y), the subgroup of G(x) fixing every vertex in {wI(x, w) or (y, w) E E(r)) is trivial.