2,141 research outputs found
Bounds on the diameter of Cayley graphs of the symmetric group
In this paper we are concerned with the conjecture that, for any set of
generators S of the symmetric group of degree n, the word length in terms of S
of every permutation is bounded above by a polynomial of n. We prove this
conjecture for sets of generators containing a permutation fixing at least 37%
of the points.Comment: 17 pages, 6 table
Large butterfly Cayley graphs and digraphs
We present families of large undirected and directed Cayley graphs whose
construction is related to butterfly networks. One approach yields, for every
large and for values of taken from a large interval, the largest known
Cayley graphs and digraphs of diameter and degree . Another method
yields, for sufficiently large and infinitely many values of , Cayley
graphs and digraphs of diameter and degree whose order is exponentially
larger in than any previously constructed. In the directed case, these are
within a linear factor in of the Moore bound.Comment: 7 page
Distance-regular Cayley graphs with small valency
We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most , the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency , and the Cayley graphs among all
distance-regular graphs with girth and valency or . We obtain that
the incidence graphs of Desarguesian affine planes minus a parallel class of
lines are Cayley graphs. We show that the incidence graphs of the known
generalized hexagons are not Cayley graphs, and neither are some other
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.Comment: 19 pages, 4 table
Metric intersection problems in Cayley graphs and the Stirling recursion
In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of
elements of order 2 and let \Gamma be the Cayley graph whose vertex set is the
group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is
determined by H. We are interested in the metric structure of this graph. In
particular, for g\in G let B_{r}(g) be the metric ball in \Gamma of radius r
and centre g. We show that the intersection numbers \Phi(\Gamma; r,
g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and
r. The results are motivated by the study of error graphs and related
reconstruction problems.Comment: 18 page
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