Louis V. Quintas

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.

Extremal f-trees and embedding spaces for molecular graphs

Abstract Problems concerning embedding trees in lattice-graph or Euclidean spaces are considered. A tree is defined to be ‘almost-embeddable’ in a lattice-graph if a sequence derived from the distance degree sequence of the lattice-graph and a corresponding sequence for the tree satisfy a specified inequality. This inequality is such that every tree that is embeddable in the lattice-graph is in the set of almost-embeddable trees. For Euclidean space embeddings the lattice-graph sequence is replaced by a sequence defined in terms of sphere packing numbers. This work has two practical objectives: Firstly, to furnish a framework within which intuitive chemical and physical notions about embedding spaces can be made explicit and self-consistent. Secondly, to obtain useable criteria which will exclude from statistical mechanical averaging procedures those molecular species which are inconsistent with a postulated embedding space. The inequality proposed here meets these objectives for molecular trees and its implications for chemical and physical theory are discussed in some detail.

Random recursive forests

A random recursive forest is defined as a union of random recursive trees. We find the expected number of trees in the uniform random recursive forest as well as the number of vertices of given degree, the maximum degree, the height of vertices, the order of branches, the root of the component containing a given vertex, and the last root of such forests.

The least number of edges for graphs having symmetric automorphism group

Abstract The least (and greatest) number of edges realizable by a graph having n vertices and automorphism group isomorphic to S m , the symmetric group of degree m , is determined for all admissible n .

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

A stability theorem for minimum edge graphs with given abstract automorphism group

Given a finite abstract group ?, whenever n is sufficiently large there exist graphs with n vertices and automorphism group isomorphic to 0. Let e (0, n) denote the minimum number of edges possible in such a graph. It is shown that for each ? there always exists a graph M such that for n sufficiently large, e(0, n) is attained by adding to M a standard maximal component asymmetric forest. A characterization of the graph M is given, a formula for e (0, n) is obtained (for large n), and the minimum edge problem is re-examined in the light of these results.

The theorem on planar graphs

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.

The Random f-Graph Process

Abstract Starting with n vertices and no edges, sequentially introduce edges so as to obtain a sequence of graphs each having no vertex of degree greater than f. The latter are called f-graphs. At each step the edge to be added is selected with equal probability from among those edges whose addition would not violate the f-degree restriction. A terminal graph of this procedure is called a sequentially generated random edge maximal f-graph and the procedure the random f-graph process of order n. This simple generalization of the classic Erdős-Renyi random graph process leads to some challenging mathematical problems and is a process related to a variety of physical applications.

The sequential generation of random f-graphs. Line maximal 2-, 3- and 4-graphs

Abstract A graph with no point of degree greater than f is called an f -graph. An f -graph to which no line can be added without introducing a point of degree greater than f is called a line maximal f-graph . We consider the following procedure. Starting with n labeled points and no lines, sequentially add lines one at a time to these points so as to obtain at each step a labeled f -graph. At each step the line to be added is selected with equiprobability from among those lines whose addition would not violate the f -degrees restriction. A terminal graph of this procedure is a random sequentially generated line maximal f -graph. Let P ( m = i ; n ; f ) denote the probability that a random sequentially generated line maximal f -graph has i points of degree less than f . The exact determination of P ( m = i ; n ; f ) is an open problem posed by P. Erdos. In this paper we obtain, for f = 2, 3 and 4, approximation functions for P ( m = i ; n ; f ) and conjecture values for lim P ( m = i ; n ; f ) as n → ∞. Heretofore, neither theoretical nor qualitative results concerning the asymptotic distribution of P ( m = i ; n ; f ) were available.