On the characteristic polynomials of spirographs and related graphs

General recursive techniques are used to determine recurrence relations for the characteristic polynomials of graphs associated with various ring compounds.     

On the maximal index of graphs with a prescribed number of edges

Among the graphs with a prescribed number of edges, those with maximal index are determined. The result confirms a conjecture of Brualdi and Hoffman.     

Spectral upper bounds for the order of a k-regular induced subgraph

Let G be a simple graph with least eigenvalue λ and let S be a set of vertices in G which induce a subgraph with mean degree k. We use a quadratic programming technique in conjunction with the main angles of G to establish an upper bound of the form |S|?inf{(k+t)q_G(t):t>-λ} where q_G is a rational function determined by the spectra of G and its complement. In the case k=0 we obtain improved bounds for the independence number of various benchmark graphs.     

The rate of molecular collisions in gases composed of ‘soft’ molecules

Equations are derived for the number of collisions per unit time in which the centres of the molecules approach to some arbitrary distance, d, or less, for molecules with spherical intermolecular potentials. The collision rates based upon this criterion of closeness-of-approach have a different dependence on temperature from those based upon cross sections determined from the viscosity. It is suggested that the rates calculated in this paper are the more appropriate for the discussion of experiments on the exchange of molecular energy and on the kinetics of chemical reactions.     

On the Multiplicities of Graph Eigenvalues

Star complements and associated quadratic functions are usedto obtain a sharp upper bound for the order of a graph withan eigenspace of prescribed codimension. It is shown that forregular graphs the bound can be reduced by 1, and that thisreduced bound is attained by a regular graph G if and only ifG is an extremal strongly regular graph. 2000 Mathematics SubjectClassification 05C50.     

Dominating Sets and Eigenvalues of Graphs

The paper is concerned with dominating sets which arise naturallyfrom a star partition of a finite graph. Techniques from linearalgebra are used to construct certain graphs in which a dominatingset is minimal.     

Tricylic hamiltonian graphs with minimal index

The index of a graph is the largest eigenvalue of an adjacency matrix whose entries are the real numbers 0 and 1. Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, those graphs with minimal index are determined.     

Graphs With Least Eigenvalue —2: A New Proof of the 31 Forbidden Subgraphs Theorem

Generalized line graphs were introduced by Hoffman Proc. Calgary Internat. Conf. on Combinatorial Structures and their applications, Gordon and Breach, New York (1970); they were characterized in 1980 by a collection of 31 forbidden induced subgraphs, obtained independently by Cvetkovi et al., Comptes Rendus Math. Rep. Acad. Sci. Canada (1980) and S. B. Rao et al., Proc. Second Symp., Indian Statistical Institute, Calcutta, Lecture Notes in Math., (1981). Here a short new proof of this characterization theorem is given, based on an edge-colouring technique.     

The penetrable sphere and related models of liquid—vapour equilibrium

The equation of state of the penetrable sphere model of liquid—vapour equilibrium is calculated by three different sequences of approximations; the first is based on the virial expansion of the equivalent two-component model in powers of the densities, the second on expansion in powers of the activity, and the third on a cumulant expansion of the configurational energy in powers of the reciprocal temperature. These sequences are examined both with the inclusion of all coefficients and with the sub-sets of coefficients appropriate to the first and second Percus-Yevick (PY) approximations. The first PY approximation gives a classical critical point whose density and temperature are accurately determined. The second PY and the complete set of coefficients yield badly-behaved series from which few conclusions can be drawn.

The penetrable sphere model is generalized to a wider class of potentials and one of these, in which the configurational energy is expressed in terms of gaussian functions is related to a two-component model of Helfand and Stillinger. It is more tractable than the original model and is examined by the same sequences of approximations. They have shown that the complete series leads to a non-classical critical point in their version of the model; here we show that the first PY approximation is classical but the second nonclassical.