34 research outputs found
Locally arc-transitive graphs of valence with trivial edge kernel
In this paper we consider connected locally -arc-transitive graphs with
vertices of valence 3 and 4, such that the kernel of the action
of an edge-stabiliser on the neighourhood is
trivial. We find nineteen finitely presented groups with the property that any
such group is a quotient of one of these groups. As an application, we
enumerate all connected locally arc-transitive graphs of valence on at
most 350 vertices whose automorphism group contains a locally arc-transitive
subgroup with
Families of Small Regular Graphs of Girth 5
In this paper we obtain --regular graphs of girth 5 with fewer
vertices than previously known ones for and for any prime performing operations of reductions and amalgams on the Levi graph of
an elliptic semiplane of type . We also obtain a 13-regular graph of
girth 5 on 236 vertices from using the same technique
Semisymmetric cubic graphs of twice odd order
The groups which can act semisymmetrically on a cubic graph of twice odd
order are determined modulo a normal subgroup which acts semiregularly on the
vertices of the graph
A characterisation of weakly locally projective amalgams related to and the sporadic simple groups and
A simple undirected graph is weakly -locally projective, for a group of
automorphisms , if for each vertex , the stabiliser induces on the
set of vertices adjacent to a doubly transitive action with socle the
projective group for an integer and a prime power .
It is -locally projective if in addition is vertex transitive. A theorem
of Trofimov reduces the classification of the -locally projective graphs to
the case where the distance factors are as in one of the known examples.
Although an analogue of Trofimov's result is not yet available for weakly
locally projective graphs, we would like to begin a program of characterising
some of the remarkable examples. We show that if a graph is weakly locally
projective with each and or , and if the distance factors
are as in the examples arising from the rank 3 tilde geometries of the groups
and , then up to isomorphism there are exactly two possible
amalgams. Moreover, we consider an infinite family of amalgams of type
(where each and ) and prove that if
there is a unique amalgam of type and it is
unfaithful, whereas if then there are exactly four amalgams of type
, precisely two of which are faithful, namely the ones related
to and , and one other which has faithful completion