The Cosine Sequences Of Distance-Regular Graphs

Cosine sequences and the intersection diagrams of a distance-regular graph are im-portant rules of studying it.In this article we study the application of cosine sequences to the intersection diagrams with rank 1 and rank 2,and the pseudo cosine sequence satisfying certain conditions.The main results are as follows.1.Let ? be a Q-polynomial distance-regular graph with a₁=0.Using the co-sine sequence,we describe the relations among a vertex and the adjacent subsets of the intersection diagram with rank 1 and rank 2,respectively.2.Let ? be a bipartite distance-regular graph and E,F be the primitive idempotents.We give the equivalent condition that E(?)F is exactly the linear combination of three primitive idempotents using cosine sequences.3.Let ? be a distance-regular-graph with a₁?0.We get the inequivalent of tight pseudo cosine sequence,and the necessary condition for pseudo cosine sequence to be(?,?)-recurrent.