On the Q‐Polynomial Property of Bipartite Graphs Admitting a Uniform Structure

ABSTRACT Let denote a finite, connected graph with vertex set . Fix and let denote the eccentricity of . For mutually distinct scalars define a diagonal matrix as follows: for we let , where denotes the shortest path length distance function of . We say that is a dual adjacency matrix candidate of with respect to if the adjacency matrix of and satisfy for some scalars . Assume now that is uniform with respect to in the sense of Terwilliger [Coding theory and design theory, Part I, IMA Vol. Math. Appl., 20 , 193–212 (1990)]. In this paper, we give sufficient conditions on the uniform structure of , such that admits a dual adjacency matrix candidate with respect to . As an application of our results, we show that the full bipartite graphs of dual polar graphs are ‐polynomial.