Dual concepts of almost distance-regularity and the spectral excess theorem

Abstract

Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first propose two dual concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as partially distance-regular graphs. Our study focuses on finding out when almost distance-regularity leads to distance-regularity. In particular, some ‘economic’ (in the sense of minimizing the number of conditions) old and new characterizations of distance-regularity are discussed. Moreover, other characterizations based on the average intersection numbers and the recurrence coefficients are obtained. In some cases, our results can also be seen as a generalization of the so-called spectral excess theorem for distance-regular graphs.