Maximum Energy Trees With Given Maximum Degree

In history, graph theory is closely related to chemistry. A chemical structure can be conveniently represented by a graph, which is called a molecular graph or a chemical graph.In chemistry, the experimental heats from the formation of conjugated hydro-carbons have a close relation with the total 7r-electron energy, which was studied already in the pioneering days of quantum chemistry. Within the Huckel molec-ular obital (HMO) approximation, the totalπ-electron energy can be calculated by means of the formula as follows where G is a molecular graph of a conjugated hydrocarbons with n vertices andλ1,λ2,…,λn are eigenvalues of G.The left-hand side of (0.0.2) is meaningful only if the underlying graph be-longs to the restricted class of molecular graph, whereas the right-hand side is well defined for all graphs. In view of this, Gutman in 1970s put forward the definition of energy of graph for any graph G by means of Equation (0.0.2). The graph energy has for a long time failed to attract the attention of both theoret-ical chemists and mathematicians. However, somewhere around the turn of the century they did realize its value. A vigorous and world-wide mathematical re-searches of graph energy started and numerous papers were published in various journals of mathematics and chemistry.One of the fundamental questions that is encountered in the study of graph energy is which graphs have greatest and smallest E-values within a given class. The main objective of this thesis is to investigate the problem of determining the trees of maximum energy among all trees with given maximum degree.In Chapter 1, we first give the basic notations and terminology related to this thesis. Meanwhile, we give an introduction to the background of the graph energy and review some known results. At last, we present the main results of this thesis.In Chapter 2, we consider the problem of determining the maximum energy trees among all trees with two vertices of maximum degreeΔ. In 2005, Lin, Guo and Li characterized the trees with a fixed number n of vertices and fixed maximum vertex degreeΔ, having maximum energy. The trees with a given maximum vertex degreeΔand maximum E happen to be trees with a single vertex of degreeΔ. In this chapter, we offer a simple proof of this result and, in addition, we characterize the trees with maximum energy among trees having two vertices of maximum degreeΔ. We also determine the trees of maximum energy among conjugated trees with given maximum vertex degreeΔ, i.e., trees with given maximum vertex degreeΔand with a perfect matching.In Chapter 3, we investigate the problem of characterizing the maximum energy trees among the trees with one maximum and one second maximum degree vertex. We determine the trees with maximum energy among all trees having one maximum degree vertex and one second maximum degree vertex.