23,268 research outputs found
Edge-transitivity of Cayley graphs generated by transpositions
Let be a set of transpositions generating the symmetric group . The
transposition graph of is defined to be the graph with vertex set
, and with vertices and being adjacent in
whenever . In the present note, it is proved that two
transposition graphs are isomorphic if and only if the corresponding two Cayley
graphs are isomorphic. It is also proved that the transposition graph is
edge-transitive if and only if the Cayley graph is
edge-transitive
Disjoint Hamilton cycles in transposition graphs
Most network topologies that have been studied have been subgraphs of transposition graphs.
Edge-disjoint Hamilton cycles are important in network topologies for improving fault-tolerance
and distribution of messaging traffic over the network. Not much was known about edge-disjoint
Hamilton cycles in general transposition graphs until recently Hung produced a construction
of 4 edge-disjoint Hamilton cycles in the 5-dimensional transposition graph and showed how
edge-disjoint Hamilton cycles in (n + 1)-dimensional transposition graphs can be constructed
inductively from edge-disjoint Hamilton cycles in n-dimensional transposition graphs. In the
same work it was conjectured that n-dimensional transposition graphs have n β 1 edge-disjoint
Hamilton cycles for all n greater than or equal to 5. In this paper we provide an edge-labelling
for transposition graphs and, by considering known Hamilton cycles in labelled star subgraphs
of transposition graphs, are able to provide an extra edge-disjoint Hamilton cycle at the inductive
step from dimension n to n + 1, and thereby prove the conjecture
The spectra of finite 3-transposition groups
We calculate the spectrum of the diagram for each finite -transposition
group. Such graphs with a given minimal eigenvalue have occurred in the context
of compact Griess subalgebras of vertex operator algebras
Error Graphs and the Reconstruction of Elements in Groups
Packing and covering problems for metric spaces, and graphs in particular,
are of essential interest in combinatorics and coding theory. They are
formulated in terms of metric balls of vertices. We consider a new problem in
graph theory which is also based on the consideration of metric balls of
vertices, but which is distinct from the traditional packing and covering
problems. This problem is motivated by applications in information transmission
when redundancy of messages is not sufficient for their exact reconstruction,
and applications in computational biology when one wishes to restore an
evolutionary process. It can be defined as the reconstruction, or
identification, of an unknown vertex in a given graph from a minimal number of
vertices (erroneous or distorted patterns) in a metric ball of a given radius r
around the unknown vertex. For this problem it is required to find minimum
restrictions for such a reconstruction to be possible and also to find
efficient reconstruction algorithms under such minimal restrictions.
In this paper we define error graphs and investigate their basic properties.
A particular class of error graphs occurs when the vertices of the graph are
the elements of a group, and when the path metric is determined by a suitable
set of group elements. These are the undirected Cayley graphs. Of particular
interest is the transposition Cayley graph on the symmetric group which occurs
in connection with the analysis of transpositional mutations in molecular
biology. We obtain a complete solution of the above problems for the
transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200
Reconstruction of permutations distorted by single transposition errors
The reconstruction problem for permutations on elements from their
erroneous patterns which are distorted by transpositions is presented in this
paper. It is shown that for any an unknown permutation is uniquely
reconstructible from 4 distinct permutations at transposition distance at most
one from the unknown permutation. The {\it transposition distance} between two
permutations is defined as the least number of transpositions needed to
transform one into the other. The proposed approach is based on the
investigation of structural properties of a corresponding Cayley graph. In the
case of at most two transposition errors it is shown that
erroneous patterns are required in order to reconstruct an unknown permutation.
Similar results are obtained for two particular cases when permutations are
distorted by given transpositions. These results confirm some bounds for
regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200
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