118512387
Groups with a $(B,\,N)$-pair and locally transitive graphs
Let Γ be an undirected graph and G a subgroup of aut (Γ). We denote by d(x, y) the distance between two vertices x and y, by E(Γ) the edge set of Γ, by V(Γ) the vertex set of Γ, by Γ(x) the set of neighbors of the vertex x and by G(x) the permutation group induced by the stabilizer G(x) on Γ(x). For each i e N, let G^x) = {a \ a e G(y) for every y with d(x, y) < ϊ). An s-path is an ordered sequence (x0, ••-,#«) of s + 1 vertices xt with xt e jP(Xi_i) for 1 < i < s and ^ ^ xt_2 for 2 < i < s. For each vertex x, let WS(JC) be the set of s-paths (x0, , xs) with x = x0. We say that the graph Γ is locally (G, s)-transitive if for every vertex x, G(x) acts transitively on Ws(x) but not on Ws+1(x) (compare [1], [11]). If, in addition, G acts transitively on V(Γ), then Γ is called (G, s)-transitive; otherwise Γ is bipartite with vertex blocks Vo and Vλ and G acts transitively on both Vo and VΊ, assuming that Γ is connected and s > 1. Now let G be a finite group with a (B, iV)-pair whose Weyl group is a dihedral group Z)2w of order 2n (n > 2) and .Γ be the incidence graph of the associated coset geometry as defined in [3, p. 129] (or [2, (15. 5. 1)]). The graph Γ has the following properties: (A) V(Γ) = Vo U V1 with Vo Π V1 0 and Γ(x) c F ^ for every vertex x e ^ (ί = 0 and 1). For j = 0 and 1 there exists a dt e N such that \Γ(x)\ = dt + 1 for every vertex xe Vi% The diameter of Γ is πand the girth 2n. (B) Γ is locally (G, τι + l)-transitive. A generalized n-gon of order (d0, dλ) is, by definition, an incidence structure whose incidence graph has the properties listed in (A). W. Feit and G. Higman have shown in [3] that finite generalized ra-gons of order (d0, dλ) with dod1 > 1 exist only for n = 2, 3, 4, 6, 8 and 12, that
46019521
The nonexistence of 8-transitive graphs
We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.
124466085
On a theorem of Goldschmidt
125321998
A characterization of the group Co 3 as a transitive extension of HS
119821589
Distance-transitive graphs and generalized polygons
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
120974879
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.
6645734
A Geometric Characterization of Certain Groups of Lie Type
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 [12] we proved the following result.
19510229
Bifurcation in difference approximations to two-point boundary value problems
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.
53988483
Permutation groups with projective unitary subconstituents
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.
122289259
On the Approximation of Fixed Points of Nonlinear Compact Operators
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.
120998826
Symmetric graphs with projective subconstituents
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 [8], it was shown that s 3. We prove the following partial result: