Maximum exponent of boolean circulant matrices with constant number of nonzero entries in its generating vector

It is well-known that the maximum exponent that an n-by-n boolean primitive circulant matrix can attain is n− 1. In this paper, we find the maximum exponent that n-by-n boolean primitive circulant matrices with constant number of nonzero entries in its generating vector can attain. We also give matrices attaining such exponents. Solving this problem we also solve two equivalent problems: 1) find the maximum exponent attained by primitive Cayley digraphs on a cyclic group whose vertices have constant outdegree; 2) determine the maximum order of basis for Zn with fixed cardinality. ∗Supported by a Faculty Career Development Award granted by UCSB in Summer 2008 and supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant MTM2006-06671. †This work was done within the activities of Centro de Estruturas Lineares e Combinatorias da Universidade de Lisboa. ‡Supported by a Summer Undergraduate Research Fellowship granted by the College of Creative Studies at UCSB. the electronic journal of combinatorics 15 (2008), #R00 1