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Orders of Symmetric Graphs

Marston Conder (University of Auckland). Jueves 19 de enero, 12 horas
Ponente:
Cuándo 19/01/2012
de 12:00 a 13:30
Dónde Salicrup
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In this talk I will describe a number of recent discoveries about
finite symmetric graphs, with arc-transitive automorphism group.
The first is the computer-assisted determination of all such
graphs on up to $10000$ vertices. An unexpected consequence of
determining all small examples was the discovery of the currently
largest known connected trivalent graph of diameter $10$,
which is a $125$-fold cover of the Petersen graph, of order $1250$.
Also these discoveries and a new application of known theory have
shown the way towards a new approach for classifying all connected
symmetric cubic graphs of order $kp$ where $k$ is fixed and $p$ is
a variable prime.  This approach can be used to prove that for
any fixed (even) $k$ and any integer $s > 1$, there are only
finitely many $s$-arc-transitive cubic graphs of order $kp$ where
$p$ is prime.  This also provides much shorter proofs of the known
classifications of connected symmetric cubic graphs of order $2p$,
$4p$, $6p$, $8p$ and $10p$, for example.  In turn, these results
have led to some further new discoveries about the orders of
symmetric graphs of higher valence (in some recent joint work
with Caiheng Li and Primoz Potocnik).