GRAPHS WITH SUBCONSTITUENTS CONTAINING L3(p)
Let T be a finite connected undirected graph, G a vertex-transitive subgroup of aut(T), {x, y) an edge of T and G,(x, y) the subgroup of G fixing every vertex at a distance of at most i from x or y. We show that if the stabilizer Gx contains a normal subgroup inducing L}(p), p a prime, on the set of vertices adjacent to x, then G5(x, y) = 1. Let T be an undirected graph with vertex set V(T) and edge set E(T) and let G be a subgroup of aut(T). For each x G V(F), we denote by T(x) the set of vertices adjacent to x, by G(x) the stabilizer of x in G and, for each / G N, by G¡(x) the subgroup {a G G(x) | a G G(u) for all u G V(T) with 3(x, «) «s i) where 9(x, u) denotes the distance between x and u. An s-path (for 5 G N) is an (j + l)-tuple (xQ,...,xs) of vertices such that x, G iX*,^,) for 1 < i < í and x/#xj_2 for 2<i<*. Let GO0,...,je,) = G(x0)n ••• n (?(*,) and G,(x0,.. .,*,) = G,(x0) n • • • n G^Xj) for each s-path (x0, ...,xs) and each i G N. If P is a group acting on a set X a an element of P, we denote by Hx the permutation group induced by H on X and by a* the permutation dfX induced by a. In the above context, the permutation group G(x)r(-x) (for x G V(T)) is known as the subconstituent of G at x. We prove the following Theorem. Let p be an arbitrary prime. Let T be a finite undirected connected graph, {x, y) an edge of T and G a subgroup of aut(T) acting transitively on V(T) such that G(x)r(x) > L3(p) (where L3(p) is to be understood as acting on the 1 + p + p2 points of the associated projective plane). Let s be the largest integer such that G acts transitively on the set of all s-paths in T. Then 2 < s < 3, G3(x, y) = 1 if s = 2 and G5(x,y)=lifs = 3. Let T be an arbitrary finite undirected connected graph, {x, y) an arbitrary edge of T and G a subgroup of aut(T) acting transitively on V(T) such that G(x)T(x) is primitive. In [6, 7], Tutte showed that | G(x) | is bounded (in fact, that | G(x) \ divides 48) if | T(x)|= 3. Our Theorem may be seen as a step in the efforts to generalize this result to graphs of arbitrary valency. By [2,(2.3)], | Gx(x, y) \ is in general a power of some prime. From this fact it follows easily that Gx(x, y) = 1 or that G(x, y)T(x) has a nontrivial normal subgroup of prime power order. This directs Received by the editors June 9, 1980 and, in revised form, May 1, 1981. AMS (A/05) subject classifications (1970).  Secondary 20G40.

title: A characterization of the group Co 3 as a transitive extension of HS

title: Distance-transitive graphs and generalized polygons
abstract: On etablit un theoreme sur un graphe Γ connexe, regulier de valence k≥3, circonference finie g, avec s≥4. Γ est (G,s)-transitif pour G≤aut(Γ) tel que G(x) est fini pour chaque sommet x

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: Presentations for (G,s)-transitive graphs of small valency
abstract: 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: On a theorem of Goldschmidt

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: The application of implicit Runge-Kutta and collection methods to boundary-value problems
abstract: The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadra- ture formula plus one. The efficient implementation of the methods is discussed and numerical examples are given. 1.  We investigate the application of certain implicit Runge- Kutta methods (cf. Butcher (2)) to the numerical solution of nonlinear boundary- value problems of the form

title: Product Integration for the Generalized Abel Equation
abstract: The solution of the generalized Abel integral equation g(t) f {k(t, s)/(t s)a}f (s) ds, 0 < a < 1, where k(t, s) is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for a E [al, 1) with al 0.2117. This interval contains the important case a = 2. Convergence of order two for a E (0, al) is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.

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: An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems
abstract: Stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o.d.e.'s without turning points.