9 research outputs found
On locally grid graphs
We investigate locally grid graphs, that is, graphs in which the
neighbourhood of any vertex is the Cartesian product of two complete graphs on
vertices. We consider the subclass of these graphs for which each pair of
vertices at distance two is joined by sufficiently many paths of length .
The number of such paths is known to be at most by previous work of
Blokhuis and Brouwer. We show that if each distance two pair is joined by at
least paths of length then the diameter is bounded by ,
while if each pair is joined by at least such paths then the diameter
is at most and we give a tight upper bound on the order of the graphs. We
show that graphs meeting this upper bound are distance-regular antipodal covers
of complete graphs. We exhibit an infinite family of such graphs which are
locally grid for odd prime powers , and apply these results to
locally grid graphs to obtain a classification for the case where
either all -graphs have order at least or all -graphs have order
for some constant
Block-transitive 2-designs with a chain of imprimitive partitions
More than years ago, Delandtsheer and Doyen showed that the automorphism
group of a block-transitive -design, with blocks of size , could leave
invariant a nontrivial point-partition, but only if the number of points was
bounded in terms of . Since then examples have been found where there are
two nontrivial point partitions, either forming a chain of partitions, or
forming a grid structure on the point set. We show, by construction of infinite
families of designs, that there is no limit on the length of a chain of
invariant point partitions for a block-transitive -design. We introduce the
notion of an `array' of a set of points which describes how the set interacts
with parts of the various partitions, and we obtain necessary and sufficient
conditions in terms of the `array' of a point set, relative to a partition
chain, for it to be a block of such a design
On the Girth and Diameter of Generalized Johnson Graphs
Let v \u3e k \u3e i be non-negative integers. The generalized Johnson graph, J(v,k,i), is the graph whose vertices are the k-subsets of a v-set, where vertices A and B are adjacent whenever |A∩B|= i. In this article, we derive general formulas for the girth and diameter of J(v,k,i). Additionally, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection