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title
Random Permutations
description
Sean Eberhard's mathematics blog
site name
Random Permutations
author
updated
2026-02-18 09:04:07
raw text
Random Permutations – Sean Eberhard's mathematics blog Skip to content Random Permutations Sean Eberhard's mathematics blog Menu Blog About me Resources An infinite sequence of triangle-free K_{2,7}-free graphs of diameter 2 Recall that a Moore graph (of diameter 2) is a regular graph of girth and diameter . Equivalently, it is a triangle-free graph such that any two nonadjacent vertices have a unique common neighbour. There are famously few Moore graphs: only the 5-cycle of degree 2, the Petersen graph of degree 3, and the Hoffman–Singleton graph of degree 7, and possibly a graph (or even several graphs) of degree and order (the “missing Moore graph”). This is the Hoffman–Singleton theorem. The Hoffman–Singleton theorem being too mysterious, it is tempting to try to find a weakening of the definition of a Moore graph that allows a few more graphs but still finitely many. One hopes that the “finitely many” part would have some more direct comb...
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