Distinguishing Vertices of Random Graphs


The distance sequence of a vertex x of a graph is (di(x))n1 where di(x) is the number of vertices at distance i from x. The paper investigates under what condition it is true that almost every graph of a probability space is such that its vertices are uniquely determined by an initial segment of the distance sequence. In particular, it is shown that for r ≥ 3 and ɛ > O almost every labelled r-regular graph is such that every vertex x is uniquely determined by (di (x))u1, where Furthermore, the paper contains an entirely combinatorial proof of a theorem of Wright [10] about the number of unlabelled graphs of a given size. © 1982, North-Holland Publishing Company

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North-Holland Mathematics Studies