We give the first polynomial-time algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give the first polynomial-time algorithm for the following related problem: Given a molecular formula, select, uniformly at random, a structural isomer having the given formula.

Computer Chemistry is of quickly increasing importance, in particular since the flood of data is rapidly growing with the introduction of Combinatorial Chemistry using methods of synthesis in large quantities (libraries) and high throughput screening. The accompanying software that allows to optimize such experiments in advance and to economize the cost of measurement afterwards uses in particular mathematical models of molecules and their description. Such models will be described here and the present state of the generation of molecular models in the computer will be discussed, from the mathematical point of view. A brief description of several applications is given. 1

MOLGEN is a computer program system which is designed for generating molecular graphs fast, redundancy free and exhaustively. In the present paper we describe its basic features, new features of the current release MOLGEN 3.5, and future developments which provide considerable improvements and extensions. 1 Introduction MOLGEN [1--7] is a generator for molecular graphs (=connectivity isomers or constitutional formulae) allowing to generate all isomers that correspond to a given molecular formula and (optional) further conditions like prescribed and forbidden substructures, ring sizes etc. The input consists of ffl the empirical formula, together with ffl an optional list of macroatoms, which means prescribed substructures that must not overlap, ffl an optional goodlist, that consists of prescribed substructures which may overlap, ffl an optional badlist, containing forbidden substructures, ffl an optional interval for the minimal and maximal size of rings, ffl an optional num...

Introduction MOLGEN[2, 3, 4] is a software package for the fast and redundancy free generation of structural formulae corresponding to prescribed data such as molecular formula, ring sizes, required or forbidden substructures, hydrogen distribution, hybridization etc. A particular version, MOLGEN-COMB[5], generates combinatorial libraries, starting from a given central molecule. The generation of a library corresponding to a given Markush formula, as encountered in chemistry patents is a quite similar task, and so MOLGEN applies to this problem as well. 2 The Comparison of two Markush Formulae Consider two Markush formulae, say the following, taken from [1] (CH 2 ) OH Cl R1 R2 R3 m R1 : methyl or ethyl (substituent variation) R2 : alkyl (homology variation) R3 : amino (position variation) m : 1-3 (frequency variation) and the following, composed by us, R5 R4 R3 R1 : methyl, ethyl, OH R2 : alkyl (1-6 carbon atoms) R3 : OH, OCH 3 , OC 2 H 5 , Methyl, Ethyl R4 : OH,

Abstract. For molecular weights up to 150, all molecular graphs corresponding to possible organic compounds made of C,H,N,O were generated using the structure generator MOLGEN. The numbers obtained were compared to the numbers of molecular graphs corresponding to actually known compounds as retrieved from the Beilstein file. The results suggest that the overwhelming majority of all organic compounds (even in this low molecular weight range) is unknown. Within the set of C6H6 isomers, a very crude and a highly sophisticated energy content calculation perform amazingly similar in predicting a particular structure’s existence as a known compound. 1.