209 research outputs found
The Bowl Championship Series: A Mathematical Review
We discuss individual components of the college football Bowl Championship
Series, compare with a simple algorithm defined by random walks on a biased
graph, attempt to predict whether the proposed changes will truly lead to
increased BCS bowl access for non-BCS schools, and conclude by arguing that the
true problem with the BCS Standings lies not in the computer algorithms, but
rather in misguided addition.Comment: 12 pages, 2 figures, submitted to Notices of the AM
The Rhodes semilattice of a biased graph
We reinterpret the Rhodes semilattices of a group
in terms of gain graphs and generalize them to all gain graphs,
both as sets of partition-potential pairs and as sets of subgraphs, and for the
latter, further to biased graphs. Based on this we propose four different
natural lattices in which the Rhodes semilattices and its generalizations are
order ideals.Comment: 7 p
Graphical representations of graphic frame matroids
A frame matroid M is graphic if there is a graph G with cycle matroid
isomorphic to M. In general, if there is one such graph, there will be many.
Zaslavsky has shown that frame matroids are precisely those having a
representation as a biased graph; this class includes graphic matroids,
bicircular matroids, and Dowling geometries. Whitney characterized which graphs
have isomorphic cycle matroids, and Matthews characterised which graphs have
isomorphic graphic bicircular matroids. In this paper, we give a
characterization of which biased graphs give rise to isomorphic graphic frame
matroids
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