Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time? We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time

Observation of long range magnetic ordering in pyrohafnate Nd2Hf2O7: A neutron diffraction study

We have investigated the physical properties of a pyrochlore hafnate Nd2Hf2O7 using ac magnetic susceptibility \chi_ac(T), dc magnetic susceptibility \chi(T), isothermal magnetization M(H) and heat capacity C_p(T) measurements, and determined the magnetic ground state by neutron powder diffraction study. An upturn is observed below 6 K in C_p(T)/T, however both C_p(T) and \chi(T) do not show any clear anomaly down to 2 K. The \chi_ac(T) shows a well pronounced anomaly indicating an antiferromagnetic transition at T_N = 0.55 K. The long range antiferromagnetic ordering is confirmed by neutron diffraction. The refinement of neutron diffraction pattern reveals an all-in/all-out antiferromagnetic structure, where for successive tetrahedra, the four Nd3+ magnetic moments point alternatively all-into or all-out-of the tetrahedron, with an ordering wavevector k = (0, 0, 0) and an ordered state magnetic moment of m = 0.62(1) \mu_B/Nd at 0.1 K. The ordered moment is strongly reduced reflecting strong quantum fluctuations in ordered state.Comment: 10 pages, 9 figures and 2 tables; to appear in Phys. Rev.

Elastic properties of graphene flakes: boundary effects and lattice vibrations

We present a calculation of the free energy, the surface free energy and the elastic constants ("Lam'e parameters" i.e, Poisson ratio, Young's modulus) of graphene flakes on the level of the density functional theory employing different standard functionals. We observe that the Lam'e parameters in small flakes can differ from the bulk values by 30% for hydrogenated zig-zag edges. The change results from the edge of the flake that compresses the interior. When including the vibrational zero point motion, we detect a decrease in the bending rigidity by ~26%. This correction is depending on the flake size, N, because the vibrational frequencies flow with growing N due to the release of the edge induced compression. We calculate Grueneisen parameters and find good agreement with previous authors.Comment: 11 pages, 12 figure

Effect of Cultivars and Season on Grafting Success in Sapota under Paschim Midnapur Conditions of West Bengal

Two sets of experiments were carried out during 2007-08 to assess incompatibility of sapota cultivars to softwood grafting, and to find out the best time for softwood grafting, in a private orchard at Jhargram of Paschim Midnapore, West Bengal. Considerable variation in success of softwood grafting among sapota cultivars was observed. Among ten cultivars studied, CO-2 showed highest compatibility with Khirnee rootstock to softwood grafting, followed by Cricket Ball and DSH-2. There was a total failure in graft-take in cultivars CO-1, DSH-1 and Guthi. Softwood grafting success was highest in sapota when carried out on 1stJuly (72%) followed by 15th&nbsp;August (70%), 5th&nbsp;June (62%) and 15th&nbsp;June (56%)