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Lower bounds on the required number of crosspoints in concentrators, a class of interconnection networks, are given. The lower bounds are obtained from a straightforward necessary condition on the number of crosspoints in sparse crossbar full capacity concentrators. Because this condition must be satisfied by all full capacity concentrators embedded in more general concentrators, the general necessary condition is established. Several sparse crossbar designs that contain the minimum number of crosspoints are presented. An extension of the results to a more general class of interconnection networks is described.