The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for twopoint resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Möbius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.

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within the experimental uncertainty. The final rise in the curve represents the zero-sound regime; in this regine, however, Eq. (3) ceases to be valid. We also would like to comment on the excellent fit that was obtained above the knee, which justifies to some degree our choice of v^ . It should be realized that a more accurate theory would have to take into account the hexagonal symmetry of the crystal as well as the quantum solid character of solid helium. These calculations are at present not available.The one outstanding difficulty in comparing Niklasson's theory with experiment is the large ratio T 1 /T 11 required. To some degree this may reflect the use of an isotropic version of the theory where proper hexagonal symmetry should have been used. We feel, in spite of this, that the effect is too large to be exclusively due to this source. We want to remark here that r 1 corresponds to a length Z = z; 2 r 1 , where v 2 is the second-sound velocity, giving the range of the coupling between local phonon flow and lattice deformations. Our results seem to indicate that An outstanding open problem in lattice statistics has been the investigation of phase transitions in Ising systems which do not possess the up-down spin-reversal symmetry. 1 * 2 A wellknown example which remains unsolved to this date is the Ising antiferromagnet in an external field. Another problem of similar nature that has been considered recently 3 " 5 is the Ising model on a triangular lattice with three-body interactions. This latter model is self-dual so that its transition temperature can be conjectured 3 ' 6 using the Kramers-Wannier argument. 7 However, the nature of the phase transition has hitherto not been known.We have succeeded in solving this model exactly. In this paper we report on our findings. It will be seen that the results are fundamentally this range is much larger in solid helium than in a quasiharmonic solid with anharmonic corrections for which the theory of Niklasson was developed.We wish to thank Rene Wanner for the initial design of the ultrasonic apparatus and for frequent discussions. . different from those of the nearest-neighbor Ising models. While the final expression of our solution is quite simple, the analysis is rather lengthy and involved. For continuity in reading, therefore, we shall first state the result. An outline of the steps leading to the solution will also be given.Consider a system of N spins a { = ±l located at the vertices of a triangular lattice L. The three spins surrounding every face interact with a three-body interaction of strength -J, so that the Hamiltonian readswith the summation extending over all faces of L. Let Z be the partition function defined by (1). We find the following expression for Z^N in the The Ising model on a triangular lattice with three-spin interactions is solved exactly. The solution, which is obtained by solving an equivalent coloring problem using the Bethe Ansatz method, is given in terms of a simple algebraic relation. The specific heat is found to diverge with in...

The number of spanning trees on a large lattice is evaluated exactly for the square, triangular and honeycomb lattices. A spanning tree of a lattice 2 is a graph drawn on 2 which connects all lattice sites and contains no polygons. For a regular lattice of N sites, the number of spanning trees on 2, TN, behaves as eZN for large N. We report here the exact values of z for the square (sQ), triangular (TR) and honeycomb (HC) lattices. More specifically let z = lim N-' In TN.

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