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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

Lipid bilayers are key components of biomembranes; they are self-assembled two-dimensional structures, primarily serving as barriers to the leakage of cell's contents. Lipid bilayers are typically charged in aqueous solution and may electrostatically interact with each other and with their environment. In this work, we investigate electrostatics of charged lipid bilayers with the main focus on the binding and bending of the bilayers. We first present a theoretical approach to charge-correlation attractions between like-charged lipid bilayers with neutralizing counterions assumed to be localized to the bilayer surface. In particular, we study the effect of nonzero ionic sizes on the attraction by treating the bilayer charges (both backbone charges and localized counterions) as forming a two-dimensional ionic fluid of hard spheres of the same diameter <i>D</i>. Using a two-dimensional Debye-H??ckel approach to this system, we examine how ion sizes influence the attraction. We find that the attraction gets stronger as surface charge densities or counterion valency increase, consistent with long-standing observations. Our results also indicate non-trivial dependence of the attraction on separations <i>h</i>: The attraction is enhanced by ion sizes for <i>h</i> ranges of physical interest, while it crosses over to the known <i>D</i>-independent universal behavior as <i>h</i> &rarr; &infin;; it remains finite as <i>h</i> &rarr; 0, as expected for a system of finite-sized ions. We also study the preferred curvature of an asymmetrically charged bilayer, in which the inner leaflet is negatively charged, while the outer one is neutral. In particular, we calculate the relaxed area difference &Delta; <i>A</i>₀ and the spontaneous curvature <i>C</i>₀ of the bilayer. We find &Delta; <i>A</i>₀ and <i>C</i>₀ are determined by the balance of a few distinct contributions: net charge repulsions, charge correlations, and the entropy associated with counterion release from the bilayer. The entropic effect is dominant for weakly charged surfaces in the presence of monovalent counterions only and tends to expand the inner leaflet, leading to negative &Delta; <i>A</i>₀ and <i>C</i>₀. In the presence of even a small concentration of divalent counterions, however, charge correlations counterbalance the entropic effect and shrink the inner leaflet, leading to positive &Delta; <i>A</i>₀ and <i>C</i>₀. We outline biological implications of our results.

Lipid bilayers are key components of biomembranes; they are self-assembled two-dimensional structures, primarily serving as barriers to the leakage of cell's contents. Lipid bilayers are typically charged in aqueous solution and may electrostatically interact with each other and with their environment. In this work, we investigate electrostatics of charged lipid bilayers with the main focus on the binding and bending of the bilayers. We first present a theoretical approach to charge-correlation attractions between like-charged lipid bilayers with neutralizing counterions assumed to be localized to the bilayer surface. In particular, we study the effect of nonzero ionic sizes on the attraction by treating the bilayer charges (both backbone charges and localized counterions) as forming a two-dimensional ionic fluid of hard spheres of the same diameter <i>D</i>. Using a two-dimensional Debye-Hückel approach to this system, we examine how ion sizes influence the attraction. We find that the attraction gets stronger as surface charge densities or counterion valency increase, consistent with long-standing observations. Our results also indicate non-trivial dependence of the attraction on separations <i>h</i>: The attraction is enhanced by ion sizes for <i>h</i> ranges of physical interest, while it crosses over to the known <i>D</i>-independent universal behavior as <i>h</i> &rarr; &infin;; it remains finite as <i>h</i> &rarr; 0, as expected for a system of finite-sized ions. We also study the preferred curvature of an asymmetrically charged bilayer, in which the inner leaflet is negatively charged, while the outer one is neutral. In particular, we calculate the relaxed area difference &Delta; <i>A</i>₀ and the spontaneous curvature <i>C</i>₀ of the bilayer. We find &Delta; <i>A</i>₀ and <i>C</i>₀ are determined by the balance of a few distinct contributions: net charge repulsions, charge correlations, and the entropy associated with counterion release from the bilayer. The entropic effect is dominant for weakly charged surfaces in the presence of monovalent counterions only and tends to expand the inner leaflet, leading to negative &Delta; <i>A</i>₀ and <i>C</i>₀. In the presence of even a small concentration of divalent counterions, however, charge correlations counterbalance the entropic effect and shrink the inner leaflet, leading to positive &Delta; <i>A</i>₀ and <i>C</i>₀. We outline biological implications of our results.

Les travaux rassemblés dans cette thèse concernent l'analyse des structures électroniques de dichalcogénures de platine et de palladium, composés solides présentant pour certains des anomalies structurales. Une étude préalable à pression ambiante est réalisée sur la famille PtQ2 (Q = O, S, Se et Te). Les quatre composés adoptent un type CdI2 polymère avec un rapport c/a très faible. A l'aide de calculs quantiques (méthodes DFT et EHTB), les phénomènes responsables sont identifiés. Un transfert électronique des anions chalcogénures vers le platine entraîne la diminution du paramètre c, tandis que des facteurs orbitalaires et géométriques provoquent l'augmentation du paramètre a. Un accent particulier est mis sur l'étude de PtO2, seul oxyde lamellaire stable. Des calculs quantiques montrent que c'est le décompte électronique qui gouverne l'arrangement structural. Le remplissage partiel des bandes d(Pt) et sp(Te) pour PtTe2 pose la question de l'équilibre des charges réel dans les chalcogénures. Pour y répondre, l'outil haute pression a été utilisé dans les cas de PtTe2, PdTe2 et PdSe2. Une analyse des distances, couplée avec des calculs DFT, montre que les formulations Pt3+(Te-1,5)2 et Pd2+(Te-1)2 semblent convenir à pression ambiante. Une très forte combinaison entre orbitales p(Q) et d(M) est constatée, empêchant l'établissement clair d'un équilibre des charges. Des expériences de synthèse et de caractérisation structurale par diffraction des rayons X mettent en évidence l'absence de transition de phase pour ces ditellurures. Dans le cas de PtTe2, l'application de pression ne conduit pas à un transfert de charges mais à un réarrangement électronique. Le composé PdSe2 subit quant à lui une transition de phase vers le type pyrite par diminution des distances Pd-Se interfeuillet. Encore une fois, il n'y a pas de transfert électronique mais un réarrangement correspondant à une distorsion Jahn-Teller coopérative. L'influence de la température est ensuite étudiée et montre des réarrangements structuraux très importants dans le type pyrite.