Boesch conjectured that there is always a uniformly-most reliable graph on n points and c edges. We present an infinite family of counterexamples to this conjecture.

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Abstract. Operators H satisfying S*H=HSwhere S is a unilateral shift on Hubert space are called Hankel operators. For a fixed shift S of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.

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