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### Abstract

Jump-number of means on graphs. (English summary) Discrete metric spaces (Marseille, 1998). European J. Combin. 21 (2000), no. 6, 767–775. Let G = (V, E) be an undirected graph, and n a positive integer. The Hamming nth-power Gn of G is the graph on V n such that (x1, x2,..., xn) and (y1, y2,..., yn) are adjacent in Gn if and only if there is at most one k ∈ {1, 2,..., n} such that xk ̸ = yk, in which case xk and yk are adjacent in G. An n-variable mapping f: V n → V is an n-mean of G if it is idempotent (i.e. it maps every constant n-tuple (a, a,..., a) to the constant a) and symmetric (i.e. it is invariant under any permutation of the variables); the jump-number of such an f is the supremum of the shortest-path distance between two vertices of the form f(x, y2,..., yn) and f(x ′, y2,..., yn) with x and x ′ adjacent in G. In the present paper, the authors give the following theorem: Given a graph g−1 n}⌉. G of girth g, the jump-number of any n-mean of G is greater than or equal to ⌈min { g 4 The proof of the theorem involves algebraic computations in the cycle-space of the graph G. Extending the result to metric spaces is discussed. In particular, it is proved that for any n-mean of a Euclidean unit circle endowed with the shortest-arc distance, there are, for every positive real ε, two n-tuples of the form (x, y2,..., yn) and (x′, y2,..., yn) with dist(x, x ′) &lt; ε, such that dist(f(x, y2,..., yn), f(x ′, y2,..., yn))&gt; min { π π

Year: 2010
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