Dusica Vidovic

Extremal Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

Quest for molecular graphs with maximal energy: a computer experiment.

If lambda(1), lambda(2),..., lambda(n) are the eigenvalues of a graph G, then the energy of G is defined as E(G) = the absolute value of lambda(1) + the absolute value of lambda(2) +.... + the absolute value of lambda(n). If G is a molecular graph, representing a conjugated hydrocarbon, then E(G) is closely related to the respective total pi-electron energy. It is not known which molecular graph with n vertices has maximal energy. With the exception of m = n - 1 and m = n, it is not known which molecular graph with n vertices and m edges has maximal energy. To come closer to the solution of this problem, and continuing an earlier study (J. Chem. Inf. Comput. Sci. 1999, 39, 984-996, ref 7), we performed a Monte Carlo-type construction of molecular (n,m)-graphs, recording those with the largest (not necessarily maximal possible) energy. The results of our search indicate that for even n the maximal-energy molecular graphs might be those possessing as many as possible six-membered cycles; for odd n such graphs seem to prefer both six- and five-membered cycles.

Coulson function and Hosoya index

Let G be a molecular graph, n the number of its vertices and φ(G,x) its characteristic polynomial. Already in 1940 Coulson expressed the total π-electron energy of conjugated unsaturated molecules in terms of the function F(x)=n−ixφ′(G,ix)/φ(G,ix). Recently, the Coulson function F(x) found applications also in modeling the structure-dependence of physico-chemical properties of alkanes. We now analyze the Coulson function and establish some of its hitherto unnoticed features, in particular its relations with the Hosoya index.

The energy of a graph and its size dependence. A Monte Carlo approach

Abstract The energy E of a graph is the sum of absolute values of graph eigenvalues and is a proper generalization of the total π -electron energy of conjugated hydrocarbons. The dependence of E on the number n of vertices and the number m of edges is examined by means of a Monte Carlo approach. It is shown that, for a fixed and sufficiently large value of n , E is, on average, an increasing function of m only for relatively small values of m and reaches a maximum for graphs in which about two-thirds of the vertex pairs are adjacent. This maximal energy increases as n 1.4 .

On Statistics of Graph Energy

Abstract The energy EG of a graph G is the sum of the absolute values of the eigenvalues of G. In the case whene G is a molecular graph, EG is closely related to the total π-electron energy of the corresponding conjugated molecule. We determine the average value of the difference between the energy of two graphs, randomly chosen from the set of all graphs with n vertices and m edges. This result provides a criterion for deciding when two (molecular) graphs are almost coeneigetic.

The Energy of a Graph and its Size Dependence. An Improved Monte Carlo Approach

Abstract In an earlier work [Gutman et al., Chem. Phys. Lett. 297, 428 (1998)] the average energy (E) of graphs with n vertices and m edges was examined, in particular its dependence on n and m . The quantity (E) was computed from a set of randomly, but not uniformly, constructed (n ,m)-graphs. We have now improved our method by constructing the (n,m)-graphs uniformly, so that every (n , m)-graph has equal probability to be generated. Differences between the old and new approaches are significant only in the case of graphs with a small number of edges.