| Let G be a distance-regular graph. Let A denote the adjacency matrix of G. Fix a vertex x of G. For each i ({dollar}rm 0le ile D),{dollar} let {dollar}rm Esb{lcub}i{rcub}sp* = Esb{lcub}i{rcub}sp*(x){dollar} denote the projection onto the {dollar}rm isp{lcub}th{rcub}{dollar} subconstituent of G with respect to x. Let T(x) denote the {dollar}doubc{dollar}-algebra generated by A and {dollar}rm{lcub}Esb{lcub}i{rcub}sp*vert 0le ile D{rcub}.{dollar} 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 {dollar}rm Esb{lcub}i{rcub}sp*W le 1{dollar} for {dollar}rm 0le ile D.{dollar} The graph G is thin if for each vertex x of G, every irreducible T(x)-module is thin.; We obtain an upper bound on the girth of a thin distance regular graph with diameter D {dollar}ge{dollar} 3 which is not a cycle, and classify those graphs where the bound is met. We also investigate the relationship between the Terwilliger algebra of an almost-bipartite graph {dollar}{lcub}cal G{rcub}{dollar} and that of its antipodal cover {dollar}{lcub}cal G{rcub}{dollar}. In particular, we show that G is thin if and only if G is thin. We investigate the structure of strongly regular graphs with {dollar}mu{dollar} = 2 that do not contain a square, where by a square, we mean a cycle x, y, z, w such that x is not adjacent to z and y is not adjacent to w. |
