Polytopes of constant weight

A study is made of the non-regular planar 3-connected graphs with constant weight.     

The complexity of finding generalized paths in tournaments

Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation υ₁υ₂…υ_n of the vertices, such that the results of the games υ₁υ₂, υ₂υ₃,…, υ_n−1υ_n match a prescribed pattern. If the pattern requires all arcs to go forwrd, i.e., υ₁ → υ₂, υ₂ → υ₃,…, υ_n−1 → υ_n, and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cn lg n, and we make the observation that O(n lg n) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-forward-backward, etc., admits an algorithm for which O(n) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn lg^αn for any α between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 4].     

Are all simple 4-polytopes Hamiltonian?

We construct an extensive family of non-Hamiltonian, 4-regular, 4-connected graphs and show that none of these graphs is the graph of a simple 4-polytope. Support from NSERC and the Math. Dept. S.F.U. is gratefully acknowledged.     

First-order formalism for dark energy and dust

This work deals with a first-order formalism for dark energy and dust in standard cosmology, for models described by a real scalar field in the presence of dust in spatially flat space. The field dynamics may be standard or tachyonic, and we show how the equations of motion can be solved by first-order differential equations. We investigate a model to illustrate how the dustlike matter may affect the cosmic evolution using this framework. PACS 98.80.Cq     

Another form of the transmission function

The transmission function describes the passage of the electric current from one point of an electric circuit to another. By now, this is also applied to molecules which are potential candidates for uses in the molecular electronics. We mean the modern branch of electronics which has a goal of reducing the sizes of its devices down to molecular ones and planning indeed to apply single molecules as conducting wires and functional components of microcircuits. For calculating the transmission function, some authors utilize the well-known idea of representing a molecule by a (molecular) graph, which allows them to apply for treating the latter also powerful methods of spectral graph theory. For instance, we refer to the paper by Fowler et al. (Chem Phys Lett. 465 2008) 142–146, where one such expression for this function is given. Our objective is to demonstrate that the same calculational result can be obtained using a different set of characteristic polynomials of graphs (which also slightly reduces a mathematical notation). Specifically, we apply one theorem of Kolmykov to the basic formula derived by these authors.     

Phased cycles

We define a phased graph G to yield an adjacency matrix A(G) having general magnitude-1 values in the same locations as the usual unphased case, but subject to the restriction that A be Hermitian. Some characteristics of phased cycles, their eigenspectra, their symmetry, and their net energy are contemplated and described.