In spite of the fact that research on the mathematical properties of the total π-electron energy E (as computed by means of the Huckel molecular orbital approximation) started already in the 1940s, many results in this area have been obtained also in the newest times. In 1978 this author published in this journal a review on E. The present article is another review on E, summarizing the progress in the theory of E, achieved since then.

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Topology and stability of conjugated hydrocarbons: The dependence of total π-electron energy on molecular topology

Sharp bounds on the distance spectral radius and the distance energy of graphs

On the minimal energy of trees with a given diameter

We extend the concept of energy to directed graphs in such a way that Coulson's integral formula remains valid. As a consequence, it is shown that the energy is increasing over the set of digraphs with n vertices and cycles of length h, with respect to a quasi-order relation. Applications to the problem of extremal values of the energy in various classes of digraphs are considered.

Energy of digraphs

For a given simple graph G, the energy of G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let P"n^@? be the unicyclic graph obtained by connecting a vertex of C"@? with a leaf of P"n"-"@?. In [G. Caporossi, D. Cvetkovic, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is C"n if n@?7 and n=9,10,11,13,15, and P"n^6 for all other values of n. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for n=4 the conjecture is not true, and P"4^3 should be the unicyclic graph with maximal energy.

Jianbin Zhang

Papers

On bicyclic graphs with maximal energy