75 research outputs found
The endomorphism type of certain bipartite graphs and a characterization of projective planes
Fan (in Southeast Asian Bull Math 25, 217-221, 2001) determines the endomorphism type of a finite projective plane. In this note we show that Fan's result actually characterizes the class of projective planes among the finite bipartite graphs of diameter three. In fact, this will follow from a generalization of Fan's theorem and its converse to all finite bipartite graphs with diameter d and girth g such that (1) d + 1 < ga parts per thousand currency sign2d, and (2) every pair of adjacent edges is contained in a circuit of length g
Abstract involutions of algebraic groups and of Kac-Moody groups
Based on the second author's thesis in this article we provide a uniform
treatment of abstract involutions of algebraic groups and of Kac-Moody groups
using twin buildings, RGD systems, and twisted involutions of Coxeter groups.
Notably we simultaneously generalize the double coset decompositions
established by Springer and by Helminck-Wang for algebraic groups and by
Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by
Devillers-Muhlherr in the context of arbitrary involutions, and we answer a
structural question on the combinatorics of involutions of twin buildings
raised by Bennett-Gramlich-Hoffman-Shpectorov
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