Spectra of unicyclic graphs

Unicyclic graphs are discussed in the context of graph orderings related to eigenvalues. Several theorems involving lexicographical ordering by spectral moments as well as the ordering by the largest eigenvalue are proved. An appendix contains a table of the 89 unicyclic graphs on eight vertices together with their spectra, spectral moments and characteristic polynomials.Carnegie Research Fellow, University of Stirling, 1985–86.     

The largest eigenvalue of a graph: A survey

This article is a survey of results concerning the largest eigenvalue (or index) of a grapn, catcgoiizeu as follows (1) inequalities lor the index, (2) graph with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications.     

Signed graphs with at most three eigenvalues

Czechoslovak Mathematical Journal - We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or...     

Star complements in regular graphs: Old and new results

We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements.     

Graphs with Least Eigenvalue −2: The Star Complement Technique

Let G be a connected graph with least eigenvalue –2, of multiplicity k. A star complement for –2 in G is an induced subgraph H = G – X such that |X| = k and –2 is not an eigenvalue of H. In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of –2. In some instances, G itself can be characterized by a star complement. If G is not a generalized line graph, G is an exceptional graph, and in this case it is shown how a star complement can be used to construct G without recourse to root systems.     

On induced matchings as star complements in regular graphs

We determine all the finite regular graphs which have an induced matching or a cocktail party graph as a star complement.     

Solutions of the Percus-Yevick and hyper-netted chain equations near the gas-liquid critical point

An intuitive approach is used to develop a theory of the transverse current correlation function in simple liquids. The memory function obtained on the basis of this approach is expressed in terms of the velocity autocorrelation function for the kinetic part and in terms of the radial distribution function, the interatomic potential and a time-dependent term which describes the relative motion of two atoms in the liquid for the potential part. The results for the memory function and the frequency spectrum are found to be in satisfactory agreement with the computer simulation studies of Levesque et al.     

The triplet distribution function in a fluid of hard spheres

The distribution functions of a fluid can be expressed as a series in powers, of the density. The coefficient of the fourth power of the density in the expansion of the triplet distribution function is obtained as a function of the separations of the molecules for a fluid of hard spheres. The coefficient is negative. It is used to estimate the amount by which the true triplet distribution function is less than the value attributed to it by the superposition approximation.     

Effective pair potentials in fluids in the presence of three-body forces. II

Earlier calculations are extended to the second order in density thus providing the non-additive contributions to the fourth virial coefficient in the presence of a weak three-body force given by the triple-dipole dispersion potential of Axilrod and Teller. We compare the Percus-Yevick approximations of Rushbrooke and Silbert and of Rowlinson at the level of the fourth virial coefficient and find that the former is more accurate. We also compare our calculations of the effective pair potential of liquid argon with the results of Mikolaj and Pings obtained from X-ray diffraction measurements.     

The theory of liquids at equilibrium

No rigorous calculation of the properties of a liquid has yet been made from a knowledge of the forces between its molecules. This review describes the method by which most solutions have been sought and the progress that has been made in the last five years.

The equilibrium properties of a simple liquid can be expressed in terms of the pair distribution function g(r), which measures the probability of finding a second molecule at distance r from a first. Kirkwood, and Born and Green showed that an approximate calculation of g(r) can be made by solving an integral equation. This equation, its solutions and their imperfections are described briefly.

More recently it has been found convenient to write the correlation function, h(r)=g(r) -1, as the sum of two terms, a direct term, c(r), and an indirect term. Four recent approximations can be described and compared by the form they assume for the direct correlation function, c(r). The most promising of these is the approximation of Percus and Yevick and its quantitative predictions are discussed in some detail.