Milan Randić

A graph theoretical approach to conjugation and resonance energies of hydrocarbons

Abstract Kekule forms of conjugated hydrocarbons are examined and an alternative description based on a set of circuits with a formal alternation of single and double bonds rather than traditional set of Kekule structures offered. The approach is graph theoretical and the analysis consists in enumeration of these unique circuits (called conjugated circuits) and subsequent parametrization when molecular properties are discussed. The concept of conjugated circuits appears close to chemical ideas and was found to directly lead to expressions for resonance energies. A previously briefly outlined scheme on selected benzenoid systems ( Chem. Phys. Letters 38 , 68 (1976)) has now been extended to larger benzenoid systems, cyclobutadiene-containing systems and several benzocyclooctatetraenes, and large number of non-alternant hydrocarbons. While in benzenoid hydrocarbons only circuits of (4n+2) size arise in cyclobutadienes also conjugated circuits of 4n type arise indicating the origin for a lesser stability of these molecules. In polycyclic non-alternant systems some compounds have only conjugated circuits of (4n+2) size, and distantly resemble benzenoids. Others have conjugated circuits of (4n+2) and 4n kind and should subsequently be expected to show a greater dissimilarity with benzenoid systems. A representative molecule of the first kind is azulene and a discrimination of non-alternants in azulenoids , i.e. those having only (4n + 2) conjugated circuits, and non-azulenoids (others) has been suggested Resonance energies of both classes of compounds are examined and the decomposition of the conjugation into circuits of different size is used as the basis for an independent estimate of resonance energies, of the quality of SCF MO calculations. An impressive agreement of the simple graph theoretical approach and fairly elaborate iterative calculations is found, which in the majority of cases is within only few hundredths of an eV. In contrast to usual empirical curve-fitting procedures the parameters involved have simple structural significance There are some formal similarities between the graph theoretical approach based on the concept of conjugated circuits and a VB semiempirical variant developed by Herndon. The congruence only reveals the combinatorial foundations of the Kekule structures on which the two approaches are based Similarities and differences (of a conceptual nature) of the two closely related schemes are discussed, and combined they offer a useful method for discussion of conjugated systems, qualitatively and quantitatively of a predictive power and accuracy associated with SCF MO calculations. Combinatorial basis of the graph theoretical scheme however offer additional interesting connections between properties, such as resonance energies, of different molecules. These follows from the expressions for resonance energies which have not been previously considered.

A novel 2-D graphical representation of DNA sequences of low degeneracy

Some 2-D and 3-D graphical representations of DNA sequences have been given by Nandy, Leong and Mogenthaler, and Randic et al., which give visual characterizations of DNA sequences. In this Letter, we introduce a novel graphical representation of DNA sequences by taking four special vectors in 2-D space to represent the four nucleic acid bases in DNA sequences, so that a DNA sequence is denoted on a plane by a successive vector sequence, which is also a directed walk on the plane. It is showed that the novel graphical representation of DNA sequences has lower degeneracy and less overlapping.

The structural origin of chromatographic retention data

Abstract Retention data on alkanes are re-examined from a structural point of view with an emphasis on the molecular connectivity. Rather than adopting empirical parameters for selected structural groupings an assumption that bond contributions depend on the formal carbon valencies in a hydrogen suppressed molecular graph is made. Except for several highly branched systems (which give problems in almost all empirical correlations) this single assumption produces a good linear correlation with the connectivity index which compares favorably with alternative schemes using ten (and more) structural empirical parameters. Furthermore, a close examination of the skeletal forms for the worst cases indicates that these are characterized by a larger number of methyl groups three bonds apart, hence by an additional assumption, that such methyl-methyl fragments cause departures from the correlation (given by the quadratic function of the number of fragments) retention indices for all alkanes considered can be well represented by a linear correlation with the connectivity index. The significance of the work is in demonstration of a dramatic reduction of the structural assumptions required. The approach should be considered as complementary to purely empirical correlations of the retention data with a large number of structural indices, pointing rather to the significance of the structural fragments involved then competing in precision with more flexible alternative schemes.

Algebraic characterization of skeletal branching

Abstract An algebraic characterization of branching of skeletal forms is proposed which is based on the concept of the comparability of functions as defined by Muirhead. The approach allows a rigorous definition of the concept of branching and suggests that structures having an identical distribution of valencies should not be discriminated. In addition situations arise when skeletal forms having a different valency configuration cannot be compared. It is discussed how this helps in clarifying ambiguities concerning branching.

A correlation between Kekulē valence structures and conjugated circuits

Abstract The recently introduced concept of conjugated circuits is re-examined and some graph theoretical properties of conjugated circuits analysed. Particularly, enumeration of conjugated circuits is extended to include disjoint conjugated circuits. It is shown that the total number of so derived conjugated circuits is always K - 1, where K is the number of Kekule valence structures for the molecule, for every Kekule valence structure. This establishes a reciprocal relation between Kekule valence structures and conjugated circuits. An interesting corollary to the theorem on the number of conjugated circuits is that one can solely from an examination of a single (any) Kekule valence structure of a molecule determine the number of Kekule valence forms and even determine the form for the remaining valence structures. Polynomials which enumerate the conjugated circuits are given for selected benzenoid hydrocarbons.

On discerning symmetry properties of graphs

Abstract A procedure is outlined which allows the symmetry properties of graphs to be systematically and rigorously investigated. It is based on a search for all the automorphisms of a graph and this is accompanied by suitably applying the procedure for recognizing identical graphs. It consists in finding all the distinctive labeling of the vertices of graph associated with the smallest binary code derived by a particular interpretation of the associated adjacency matrix. No prior cognizance of symmetry operations is required which is in contrast to the usual discussions of the symmetry properties of molecules which are based on the knowledge of pertinent symmetry operations. This is important since in graphs it is neither apparent nor generally possible to simply enlist those permutations of labels which leave the connectivity invariant (i.e., do not alter the form of the adjacency matrix). The procedure is applied to the Petersen graphs and the Desargues-Levi graph, both associated with isomerizations of trigonal bipyramidal complex and other chemical transformations. It is shown that these graphs of high symmetry belong to symmetry groups of order 120 and 240 respectively. The approach can also provide a basis for the development of the symmetry properties of non-rigid molecules in which connectivity is preserved.

On comparability of structures

Abstract The problem of comparability of structures and functions defined on structures is considered following the early work of Muirhead (confined to integral parametrization) and more modern generalization of Karamata. The basis of such comparabilities is the set of inequalities which when satisfied permit one to establish a relative order (importance) of the structures or associated functions. Chemical illustrations are given for both, Muirhead and Karamata inequalities. It is outlined how to go beyond the partial ordering in such comparison when additional information on the trends among the parameters involved is available.

On the parity of Kekulé structures

The concept of parity of Kekule structures is critically examined. While it appears to be desirable to have a discriminatory device to classify Kekule structures into those with a predominant stabilizing effect and those which destabilize a system relative to an appropriate hypothetical acyclic structure with the same number of CC double bonds, the concept of parity as introduced in the early development of valence theory is found unsatisfactory. A remedy for the situation is proposed which is based on a closer examination of the conjugated circuits present in a system. It is shown that while in many instances the new criterion gives the same classification of Kekule structures as the concept of parity, in other situations where the parity classification leads to ambiguities or unacceptable results the new scheme works well.

Canonical labeling of proteome maps.

We propose a canonical labeling of proteome maps, which enables one to sort and catalog the maps in a simple way. The canonical label of a proteome map is based on the canonical labeling of vertexes of Hasse diagram embedded in the map resulting in the adjacency matrix, the rows of which when viewed as binary numbers are the smallest possible such numbers. The use of the approach in documentation is illustrated with the proteome maps of liver cells of healthy male Fisher F344 rats and the rats treated with different peroxisome proliferators.

On the construction of endospectral graphs

The characteristic polynomial of a graph is defined as (1)” det(A xl) , where I is the identity matrix of size n and A is the adjacency matrix of a graph having n vertices. In 1957 Collatz reported different graphs having the same characteristic polynomial, the results in part based on work of Sinogowitz, who died in 1944 during the war [2]. The topic was revived by Fisher [4] who exhibited several isospectral (also called cospectral) graphs (i.e., graphs having the same spectrum or same characteristic polynomial). In the following years the number of known isospectral graphs increased considerably [l, 6, 7, 171. By 1970 Mowshowitz pointed out an infinite family of isospectral graphs [12], and later Schwenk reported that isospectral graphs are the rule rather than the exception when one considers graphs of increasing size. Examination of numerous isospectral graphs indicated that in some cases there is no apparent relationship between the pair of isospectral graphs, while in numerous other situations isospectral graphs can be simply derived from a single isospectral pair. We will refer to the former as sporadic and the latter as generic isospectral pairs, respectively. For an illustration see FIGURE 1, where examples of each are shown. Observe that “production” of additional isospectral graphs is straightforward, once the “isospectral points,” that is, the special vertices that act as the substitution sites in the generic pair, are identified [8, 91. Moreover, in some apparently rare situations the parent isospectral pair can itself be derived from a single common subgraph. In FIGURE 2 we show two cases discussed in the literature, a cyclic case and an acyclic case. Each of the graphs has two special vertices, to be referred to as endospectral points, which have the property that any substitution at those sites will produce an isospectral pair. We refer to these graphs as endospectral [14], and in this contribution we will restrict the attention to endospectral trees only. The acyclic graph of FIGURE 2 was studied by Schwenk [ 181, who proved that Characteristic polynomials of subgraphs G-u and G-v, where u, v are the endospectral points, are identical. This alone ensures the isospectral character of graphs