Cayley graphs generated by small degree polynomials over finite fields

We improve upper bounds of F. R. K. Chung and of M. Lu, D. Wan, L.-P. Wang, X.-D. Zhang on the diameter of some Cayley graphs constructed from polynomials over finite fields

On the structure of quantum automorphism groups

We compute the K-theory of quantum automorphism groups of finite dimensional C∗-algebras in the sense of Wang. The results show in particular that the C∗-algebras of functions on the quantum permutation groups S+n are pairwise non-isomorphic for different values of n. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena

A note on QUBO instances defined on Chimera graphs

McGeoch and Wang (2013) recently obtained optimal or near-optimal solutions to some quadratic unconstrained boolean optimization (QUBO) problem instances using a 439 qubit D-Wave Two quantum computing system in much less time than with the IBM ILOG CPLEX mixed-integer quadratic programming (MIQP) solver. The problems studied by McGeoch and Wang are defined on subgraphs -- with up to 439 nodes -- of Chimera graphs. We observe that after a standard reformulation of the QUBO problem as a mixed-integer linear program (MILP), the specific instances used by McGeoch and Wang can be solved to optimality with the CPLEX MILP solver in much less time than the time reported in McGeoch and Wang for the CPLEX MIQP solver. However, the solution time is still more than the time taken by the D-Wave computer in the McGeoch-Wang tests.Comment: Version 1 discussed computational results with random QUBO instances. McGeoch and Wang made an error in describing the instances they used; they did not use random QUBO instances but rather random Ising Model instances with fields (mapped to QUBO instances). The current version of the note reports on tests with the precise instances used by McGeoch and Wan