Symmetries of Graphs and Networks

Symmetric graphs of diameter 2 with complete normal quotients Carmen Amarra University of Western Australia, Australia [email protected] A graph has diameter 2 if it is not a complete graph and if every pair of nonadjacent vertices is joined by a path of length 2. Our general problem is to examine the overall structure of graphs which are both arc-transitive and of diameter 2 using normal quotients. We identify the basic graphs in the class of arc-transitive diameter 2 graphs to be those Γ which satisfy one of the following (relative to some group G of automorphisms of Γ): (i) Γ does not have a nontrivial G -normal quotient, or (ii) all nontrivial G -normal quotients of Γ are complete graphs. In this talk we discuss basic graphs which are of type (ii), in particular the case where there are at least 3 nontrivial normal quotients. For this case we can identify all the graphs that arise, using the classification of the transitive finite linear groups. Leonard triples associated with hypercubes and their antipodal quotients George Brown University of Wisconsin Madison, USA [email protected] LetA be the unital associative algebra over C with generators x , y , z and relations x y + y x = 2z , y z + z y = 2x and z x + x z = 2y . We find the finite-dimensional irreducible A modules and show that x , y , z act on these modules as bipartite or almost bipartite Leonard triples. We define an operator s on finite-dimensional sl2-modules that gives them anA structure. Let d denote a nonnegative integer and let Qd denote the graph of the d -dimensional hypercube. It is known that the Terwilliger algebra of Qd has an sl2-module structure. When d is even, we show that applying s to the Terwilliger algebra of Qd produces the Terwilliger algebra of the alternate Q-polynomial structure of Qd . When d is odd, we show that applying s to the Terwilliger algebra of Qd produces the Terwilliger algebra of the antipodal quotient of Qd .