431 research outputs found
Notes on the connectivity of Cayley coset digraphs
Hamidoune's connectivity results for hierarchical Cayley digraphs are
extended to Cayley coset digraphs and thus to arbitrary vertex transitive
digraphs. It is shown that if a Cayley coset digraph can be hierarchically
decomposed in a certain way, then it is optimally vertex connected. The results
are obtained by extending the methods used by Hamidoune. They are used to show
that cycle-prefix graphs are optimally vertex connected. This implies that
cycle-prefix graphs have good fault tolerance properties.Comment: 15 page
Diameter of Cayley graphs of permutation groups generated by transposition trees
Let be a Cayley graph of the permutation group generated by a
transposition tree on vertices. In an oft-cited paper
\cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is
shown that the diameter of the Cayley graph is bounded as
\diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n
\dist_T(i,\pi(i))}, where the maximization is over all permutations ,
denotes the number of cycles in , and \dist_T is the distance
function in . In this work, we first assess the performance (the sharpness
and strictness) of this upper bound. We show that the upper bound is sharp for
all trees of maximum diameter and also for all trees of minimum diameter, and
we exhibit some families of trees for which the bound is strict. We then show
that for every , there exists a tree on vertices, such that the
difference between the upper bound and the true diameter value is at least
.
Observe that evaluating this upper bound requires on the order of (times
a polynomial) computations. We provide an algorithm that obtains an estimate of
the diameter, but which requires only on the order of (polynomial in)
computations; furthermore, the value obtained by our algorithm is less than or
equal to the previously known diameter upper bound. This result is possible
because our algorithm works directly with the transposition tree on
vertices and does not require examining any of the permutations (only the proof
requires examining the permutations). For all families of trees examined so
far, the value computed by our algorithm happens to also be an upper
bound on the diameter, i.e.
\diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n
\dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535
New results for the degree/diameter problem
The results of computer searches for large graphs with given (small) degree
and diameter are presented. The new graphs are Cayley graphs of semidirect
products of cyclic groups and related groups. One fundamental use of our
``dense graphs'' is in the design of efficient communication network
topologies.Comment: 15 page
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