On the Terwilliger Algebra of Bipartite Distance-regular graphs (Disposition of the PhD Thesis)

This is a presentation of the PhD thesis topic.

Let G denote a bipartite distance-regular graph with diameter D ge 4 and valency k ge 3. Let X denote the vertex set of G, and let A denote the adjacency matrix of G. For x in X and for 0 le i le D, let G_i(x) denote the set of vertices in X that are distance i from vertex x. Define parameters Delta_i (1le ile D-1) in terms of the intersection numbers by Delta_i=(b_{i-1} – 1)(c_{i+1} – 1) – (c_2 – 1)p^i_{2i}. In PhD thesis we show that Delta_2 = 0 implies D le 5 or c_2 in {1,2}.

For x in X let T=T(x) denote the subalgebra of MX generated by A, Es_0, Es_1, ldots, Es_D, where for 0 le i le D, Es_i represents the projection onto the i-th subconstituent of G with respect to x. We refer to T as the Terwilliger algebra of G with respect to x. By the endpoint of an irreducible T-module W we mean min{i | Es_iW ne 0}.

Consider a bipartite DRG G with one of the following properties:

(b.1) G has, up to isomorphism, a unique irreducible T-module W of endpoint 2, this module is not thin, dim(Es_iW)le 2 for every i (2le ile D-1) and dim(Es_2W)= 1.

(b.2) Delta_i=0 for every i (2le ile D-1).

(b.3) G has the property that for 2le ile D-1, there exist complex scalars alpha_i, beta_i such that for all x,y,zin X with partial(x,y)=2, partial(x,z)=i, partial(y,z)=i, we have alpha_i+beta_i|G_1(x)capG_1(y)capG_{i-1}(z)|=|G_{i-1}(x)capG_{i-1}(y)capG_{1}(z)|.

(b.4) Delta_2=0, Delta_inot=0 for at least one i (3le ile D-2), and (b.3) holds.

In this PhD Thesis we will show that the property (b.4) implies property (b.1). Note that property (b.4) includes (b.3) by definition, and we are interested in bipartite distance-regular graphs with property (b.3) because they arise as a natural family in the study of the Terwilliger algebra of a bipartite distance-regular graph, as we will see by the following  very important example. Suppose that G is Q-polynomial. Then G has, up to isomorphism, at most one irreducible T-module of endpoint 2 and diameter D-2, at most one irreducible T-module of endpoint 2 and diameter D-4 (they are both thin), and no other irreducible T-modules of endpoint 2, see [J. S. Caughman, IV, The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete Math. 196 (1999), 65–95.]. Furthermore, Terwilliger's {it balanced set condition} ([P. Terwilliger, A new inequality for distance-regular graphs, Discrete Math. 137 (1995), 319–332.]) implies the property (b.3) ([Š. Miklavič, On bipartite Q-polynomial distance-regular graphs, European J. Combin. 28 (2007), 94–110.]).

In this PhD Thesis, we will not assume the Q-polynomial property for G, but rather the property (b.3) above, together with Delta_2=0. It is our goal to describe the irreducible T-modules with endpoint 2 for this case. We assume that Delta_inot=0 for at least one i (3le ile D-2), since graphs with property (b.2) are already well-understood ([B. Curtin, Almost 2-homogeneous bipartite distance-regular graphs, European J. Combin. 21 (2000), 865–876.]). First we will show that in case when c_2le 2, there exists a certain equitable partition of the vertex set of G which, for case c_2=1, involves 3(D-1)+ell cells and, for case c_2=2, involves 4(D-1)+2ell cells, for some integer ell with 0leellle D-2. We use this equitable partition to describe the irreducible T-modules with endpoint 2 for this case.

In the second part of the thesis we assume G is a bipartite Q-polynomial distance-regular graph with diameter D ge 4, valency k ge 3 and intersection numbers b_i, c_i. Caughman proved in [J. S. Caughman, IV, Bipartite Q-polynomial distance-regular graphs, Graphs Combin. 20 (2004), 47–57.] that if D ge 12 then G is either the D-dimensional hypercube, or the antipodal quotient of the 2D-dimensional hypercube, or the intersection numbers of G satisfy c_i = (q^i-1)/(q-1) ; (0 le i le D) for some integer q at least 2. Note that if c_2 le 2, then the last of the above possibilities cannot occur. The aim of this PhD thesis will also be to further investigate these graphs. We will show that if c_2 le 2 then G is either the D-dimensional hypercube, or the antipodal quotient of the 2D-dimensional hypercube, or D=5.

In the problems of (i) finding equitable partition for c_2le 2; (ii) describing irreducible T-modules with endpoint 2 and (iii) proving that Q-polynomial distance-regular graph G is either the D-dimensional hypercube, or the antipodal quotient of the 2D-dimensional hypercube, or D=5; we use combinatorial and linear algebra methods similar to those that are used in [Š. Miklavič and S. Penjić, "On bipartite Q-polynomial distance-regular graphs with c_2 ≤ 2", The Electronic Journal of Combinatorics 21(4) (2014), #P4.53.], [M. S. MacLeana, Š. Miklavič, S. Penjić, "On the Terwilliger algebra of bipartite distance-regular graphs with Delta_2=0 and c_2=1", Linear Algebra and its Applications 496 (2016), 307–330.] and [S. Penjić, "On the Terwilliger algebra of bipartite distance-regular graphs with Delta_2=0 and c_2=2", Discrete Mathematics 340 (2017), 452–466].