Phonon density-of-states in the new superconductor MgB 2

We provide here experimental data on the phonon density-of-states of MgB₂ obtained by the inelastic neutron scattering technique. The measurements were performed for the natural boron-based magnesium diboride with use of a time-of-flight neutron spectrometer. Several phonon bands were observed in the phonon spectrum at energies of about 33, 55, 82 and 99 meV. We show that the cut-off energy of the density-of-states occurs at around 105 meV which is much higher than expected so far from heat-capacity data and partially explains the high T _c value observed for MgB₂. The characteristic phonon energies are indicative of an intermediate coupling regime in this compound. We conclude that a much needed neutron experiment aimed at the study of the isotopic effect in the phonon density-of-states of MgB₂ is conceivable. Received 19 March 2001     

A census of semisymmetric cubic graphs on up to 768 vertices

A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K_1,3 as a normal quotient is investigated in detail. Supported in part by N.Z. Marsden Fund (grant no. UOA 124) and N.Z. Centres of Research Excellence Fund (grant no. UOA 201) Supported in part by “Ministrstvo za šolstvo, znanost in šport Slovenije”, research program no. 101-506. Supported in part by research projects no. Z1-4186-0101 and no. Z1-3124-0101. The fourth author would like to thank the University of Auckland for hospitality during his visit there in 2003.     

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex. Supported in part by the N.Z. Marsden Fund (grant no. UOA0124).     

A Classification of 2-Arc-Transitive Circulants

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of k-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.