Allen J. Schwenk

The number of caterpillars

A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed. The number of nonisomorphic caterpillars with n+4 points is 2^n + 2^[^n^2^]. This neat formula is proved in two ways: first, as a special case of an application of Polyas enumeration theorem which counts graphs with integer-weighted points; secondly, by an appropriate labeling of the lines of the caterpillar.

The distribution of degrees in a large random tree

For labeled trees, Renyi showed that the probability that an arbitrary point of a random tree has degree k approaches l/e(k-l)!. For unlabeled trees, the answer is different because the number of ways to label a given tree depends on the order of its automorphism group. Using arguments involving combinatorial enumeration and asymptotics, we evaluate the corresponding probabilities for large unlabeled trees.

INTEGRAL STARLIKE TREES

In this note we determine which of the trees homeomorphic to a star have a spectrum consisting entirely of integers. We also specify the integral double stars, and we consider the problem of trees with more complicated structure. Subject classification (Amer. Math. Soc. (MOS) 1970) : 05 C 05.

The communication problem on graphs and digraphs

The communication problem discussed in this article has been given various names in the literature. For example, Baker and Shostak (1) call it the “gossip problem” while Hajnal et al. (3) refer to it as the “telephone disease”. This is analogous to the well-known term “the four color disease” [(4), p. 1261, which arose from the fact that the problem has so many features of an ailment, the main difference being that this one was settled so effortlessly, relatively speaking. In the literature of psychology the “common symbol problem” (2) is equivalent to it, although in disguise. Recently, Moon (5) studied a probabilistic variation of the communication problem in which the calls are made at random.

THE CONSTRUCTION OF COSPECTRAL COMPOSITE GRAPHS

An operation loosely described as a type of composition of graphs is studied. Under rather flexible conditions, the resulting composite graphs must be cospectral. This operation is sufficiently powerful to generate eighty‐one cospectral pairs with at most nine vertices. These pairs include the unique smallest cospectral pair, the smallest cospectral connected pair, and one pair of trees with nine vertices. It is felt that this operation provides a unified explanation of cospectrality in several cases that were previously viewed as coincidental.

Enumeration of graphs with signed points and lines

Our object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs. We also treat several associated problems for which these configurations are self-dual with respect to sign change. We find that the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group. This counting technique is based on Polyas Enumeration Theorem and the Power Group Enumeration Theorem. Using a suitable computer program, we list the number of graphs of each type considered up to twelve points. Sharp asymptotic estimates are also obtained.

An asymptotic evaluation of the cycle index of a symmetric group

Let Z(S, ; f(x )) denote the polynomial obtained from the cycle Index of the symmetric group Z(S,) by replacing each variable s, by f{x’), Let f(x) have a Taylor series with radius of convergence p of the form f(x)= .Y’ + a,,,~~*‘+ ak,,:xk~‘+ a.. <with every a, 20. F,inally, Itr o< x < 1 and let x G p. We prove that This limit is used to estimate the probabihty (far n and p both large} that a point [chosen at random from a random p-point tree has degree n + 1. These limitilng probabilities are independent of p and decrease geometrically in n. contrasring with the labe!ecl limiting probabilities of l/n!e. In order to prove the main the<?rem, an appealing generalization o/r the principle of inclusion and exclusion is presented. 1. htroductiou In the paper [(u] we examined the probability that a point chosen at random in a large random tree has degree k = n -F- 1. Following the notation of Hararr [2] and of Harary and Palmer 131, we let t(x) and L?(x) denote the generating functions for ordinary and nvoted trees. These functions have the common radius of convergence p, Similarly, we defined G!‘“‘(X) and D’“‘(x) to be the generating functions for the total number o.f points with degree k amcjng all ordinary and all planted trees. We then found an asymptotic formula for the probability that a point has degree k :

Pseudosimilar vertices in a graph

Abstract : Dissimilar vertices whose removal leaves isomorphic subgraphs are called pseudosimilar. We construct infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex. Potential impact on the Reconstruction Conjecture is considered. We also construct, for each n, graphs containing a subset of vertices of size n which are mutually pseudosimilar. The analogous problem for mutually pseudosimilar edges is introduced. (Author)

On unimodal sequences of graphical invariants

Let bi denote the number of ways to select a subset of i independent edges in a given graph. It is shown that the sequence of bis is unimodal, that is, there exists an r such that b0 … > bm. Similarly, for any bigraph, the nonzero coefficients in the characteristic polynomial are shown to be unimodal in magnitude. Finally, it is suggested that the approach used here might be applied to verify the conjecture that the coefficients in the chromatic polynomial are unimodal in magnitude.

Acquaintance Graph Party Problem

(1972). Acquaintance Graph Party Problem. The American Mathematical Monthly: Vol. 79, No. 10, pp. 1113-1117.