#### Keywords

Chemistry -- Mathematics, Eigenvalues, Graph theory, Huckel molecular orbitals

#### Abstract

The energy of a graph began with German physicist, Erich H¨uckel’s 1931 paper, Quantenttheoretische Beitr¨age zum Benzolproblem. His work developed a method for computing the binding energy of the π-electrons for a certain class of organic molecules. The vertices of the graph represented the carbon atoms while the single edge between each pair of distinct vertices represented the hydrogen bonds between the carbon atoms. In turn, the chemical graphs were represented by an n × n matrix used in solving Schr¨odinger’s eigenvalue/eigenvector equation. The sum of the absolute values of these graph eigenvalues represented the total π-electron energy. The criteria for constructing these chemical graphs and the chemical interpretations of all the quantities involved made up the H¨uckel Molecular Orbital theory or HMO theory. In this paper, we will show how the chemical interpretation of H¨uckel’s graph energy evolved to a mathematical interpretation of graph energy that Ivan Gutman provided for us in his famous 1978 definition of the energy of a graph. Next, we will present Charles Coulson’s 1940 theorem that expresses the energy of a graph as a contour integral and prove some of its corollaries. These corollaries allow us to order the energies of acyclic and bipartite graphs by the coefficients of their characteristic polynomial. Following Coulson’s theorem and its corollaries we will look at McClelland’s first theorem on the bounds for the energy of a graph. In the corollaries that follow McClelland’s 1971 theorem, we will prove the corollaries that show a direct variation between the energy of a graph and the number of its vertices and edges. Finally, we will see how this relationship led to Gutman’s conjecture that the complete graph on n vertices has maximal energy. Although this was disproved by Chris Godsil in 1981, we will provide an independent counterexample with the help of the software, Maple 13

#### Notes

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#### Graduation Date

2011

#### Semester

Fall

#### Advisor

Mohapatra, Ram

#### Degree

Master of Science (M.S.)

#### College

College of Sciences

#### Department

Mathematics

#### Degree Program

Mathematical Science

#### Format

application/pdf

#### Identifier

CFE0004184

#### URL

http://purl.fcla.edu/fcla/etd/CFE0004184

#### Language

English

#### Release Date

December 2014

#### Length of Campus-only Access

3 years

#### Access Status

Masters Thesis (Open Access)

#### Subjects

Dissertations, Academic -- Sciences, Sciences -- Dissertations, Academic

#### STARS Citation

Zimmerman, Steven, "Hückel Energy Of A Graph: Its Evolution From Quantum Chemistry To Mathematics" (2011). *Electronic Theses and Dissertations*. 1736.

http://stars.library.ucf.edu/etd/1736