John W. Kennedy

Wiener numbers of random benzenoid chains

Abstract An explicit analytical expression for E ( W ( n )), the expected value of the Wiener number of a random benzenoid chain with n hexagons is obtained. In the general case, E ( W ( n )) is not a polynomial in the variable n .

Perfect matchings in random hexagonal chain graphs

Simple exact formulae are obtained for the expected value of the number of perfect matchings in a random hexagonal chain and for the asymptotic behavior of this expectation.

A characterization of graphs of diameter two

On montre que les graphes de diametre 2 peuvent etre caracterises par une simple condition portant sur leurs complementaires

Probability models for random f -graphs

In this paper we use the term f-graph to mean a simple graph (possibly not connected) in which no point has degree greater than some fixed number,

The theorem on planar graphs

Our motivation for the study of such graphs stems from their importance as models for physical systems (see, e.g., [ t , 3, 4, 10, 19, 21, 251). In these systems degree limitations are often a natural consequence of the physics involved. For example, in graph-like chemical models, where points represent atoms, the degree restrictions are a consequence of chemical “valencies.” We detail here probability models for randomly constructedf-graphs, or random f-graphs as we shall call them (see the following four sections). The analysis of these structures shows that they bear many similarities to the classic random graph model of Erdos and Renyi [6, 7, 241, for which there are no degree restrictions. Unlike Erdos-Renyi random graphs, which have been extensively studied by mathematicians, their f-graph counterparts have been largely neglected by the mathematical community and have remained in the province of chemists and physicists. This is strange since random f-graphs are more general in the sense that the Erdos-Renyi model is a special case of randomf-graphs reached in the limit asf tends to infinity [17], [18]. In another sense it is less strange in that the analysis of randomfgraphs is more challenging. Of particular interest to us is the course charted during the evolution of random Jlgraphs [l], [19] (see the sections on induced subgraphs and randomly pointweighted random fgraphs). Here there are many new and interesting problems to be found that result entirely from the imposition of degree restrictions in the construction of random graphs. The concept of randomly point-weighted random fgraphs, hinted at in [16] is briefly mentioned. We use the binomial probability distribution frequently; thus, it is convenient to use the notation:

Asymptotic number of symmetries in locally restricted trees

Abstract In the late 1920s several mathematicians were on the verge of discovering a theorem for characterizing planar graphs. The proof of such a theorem was published in 1930 by Kazimierz Kuratowski, and soon thereafter the theorem was referred to as the Kuratowski Theorem. It has since become the most frequently cited result in graph theory. Recently, the name of Pontryagin has been coupled with that of Kuratowski when identifying this result. The events related to this development are examined with the object of determining to whom and in what proportion the credit should be given for the discovery of this theorem.

Antigenesis: a cascade-theoretical anaylsis of the size distributions of anti-gen-antibody complexes

Abstract Exact and asymptotic formulas were used to compute the number of symmetries in several types of unlabeled trees with vertices of small degree.

The Long and the Short of Cascade Trees: Height and a Duality in Cascade Processes a

Abstract Various ad hoc approaches have been employed for the calculation of statistical properties of antigen—antibody complex systems. These approaches, however, overlook the fact that a systematic approach to such problems is available within the scope of the class of random graph processes known as branching or cascade processes. In this paper, we introduce, with sufficient detail for the reader to follow, the details of a cascade theoretical model for antigen—antibody systems and show how the molecular weight distribution for a particular system may be explored. The advantage of this systematic approach lies both in the ease with which statistical quantities can be computed once the model is set up, and the facility with which any simplifying assumptions of the model can be described and possibly relaxed should the need to do so arise.

Counting and coding identity trees with fixed diameter and bounded degree

Random cascade [S, 91 or branching 114,231 processes form a class of probability models with wide applicability (see, for example, [3, 10, 15, 221). Our interest in them stems from their applications in chemistry and physics, and because of the close connection between these random processes and the theory of random graphs [15, 161. In this paper we deal with cascade processes. We review their key features in the language of random rooted trees and comment on the important, but unsolved, problem of tree height. We deal in detail with the threshold beyond which a cascade may continue indefinitely and the distribution of finite-order trees thereafter, when cascades are statistically heterogeneous. This leads to a useful duality relation between preand postthreshold cascades that can be exploited to deal with questions of theoretical and scientific origin. We make use of the following graph theoretic terminology. A free is a connected graph with no cycle. Arooted tree is a tree with one vertex distinguished; this vertex is called the root. Every vertex in a rooted tree lies on a unique path connecting it with the root. Thus, the distance of a vertex u from the root in a rooted tree T is the number of edges in the unique path connecting Y with the root. This distance we call the root-distance of v in T. The probability distribution of a discrete random variable, 2, is conveniently dealt with in terms of itsprohahiZity~eneratiltgfunction (pgf). A pgf is a formal power series in a variable, t , such that the coefficient of tJ is P(Z = j ) , the probability that 2 takes the value j.

The number of alkanes having n carbons and a longest chain of length d: An application of a theorem of Polya

Abstract Identity trees with bounded maximum degree play a fundamental role in applications-oriented problems, especially when the trees are classified by their diameters. This paper offers results related to enumeration of such tree classes obtained by extending the methods of Gordon and Kennedy [The counting and coding of trees of fixed diameter, SIAM J. Appl. Math. 28 376–398 (1975)]. We set our results into the context of other enumerative work on identity trees. We derive formulae for the numbers of identity trees of various types, with fixed diameter and maximum degree. This then leads to asymptotic formulae (for large diameter). By combining these with formulae derived by Gordon and Kennedy [loc. cit.] we obtain the asymptotic fractions of identity trees among trees in various classes. These fractions are juxtaposed with asymptotic results that have appeared elsewhere. Our final section derives algorithms for integer coding and decoding identity trees in a way that is highly convenient for computer applications.