The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal 2-cover Original Research Article

Let G be a distance-regular graph. Let A denote the adjacency matrix of G. Fix a vertex x of G. For each i (0⩽i⩽D), let Ei∗=Ei∗(x) denote the projection onto the ith subconstituent of G with respect to x. Let T(x) denote the C-algebra generated by A and {Ei∗ | 0⩽i⩽D}. We call T(x) the Terwilliger algebra of G with respect to x. An irreducible T(x)-module W is said to be thin if dim Ei∗W⩽1 for 0⩽i⩽D. The graph G is thin if for each vertex x of G, every irreducible T(x)-module is thin. A distance-regular graph G=(X,R) with diameter D is said to be almost-bipartite if the intersection numbers satisfy ai(G)=0 (0⩽i⩽D−1) and aD(G)≠0. Let G=(X,R) be an almost-bipartite distance-regular graph with diameter D. Then there is a distance-regular graph G=(X,R) of diameter D=2D+1 which is an antipodal 2-cover, and a 2-to-1 surjection π : X→X which preserves adjacency. It is known that G and G determine each other, up to isomorphism. We investigate the relationship between the Terwilliger algebras and their module structures of two graphs related in this way. In particular, we show that G is thin if and only if G is thin.