Algorithm 546: SOLVEBLOK [F4]
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for dtrect commercial advantage, the ACM copymght notice and the title of the pubhcation and its date appear, and notme is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to repubhsh, requires a fee and/or specific permission.

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: 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: 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: 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: 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: 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.

title: GRAPHS WITH SUBCONSTITUENTS CONTAINING L3(p)
abstract: 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: Edge-symmetric graphs

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.