| Graph theory has many important applications in chemistry. Especially, there ex-ists a close correspondence between eigenvalues of graphs and the molecular orbital energy levels of π-electrons in conjugated hydrocarbons. The total π-electron energy E of the majority of conjugated hydrocarbons, under the Hu chkel molecular orbital approximation, can be computed by the following formula at the initial period of the quantum chemical theorywhere λi are eigenvalues of the molecular graph G representing the conjugated hydro-carbon. Note that right-hand side of Eq.(0.2) is well defined for all graphs. In view of this, if G is an any given graph, then by means of Eq.(0.2), we can define the energy E(G) of G.In1977, the famous mathematic-chemist Gutman first proposed the theory of graph energy which has been evolved into one of the very active directions on graph the-ory ever since. Moreover, it has been rather widely concerned by theoretical chemists and mathematicians. Particularly since2000, graph energy has been deeply developed in which many important results have been discovered. As we know, quasi-order rela-tion is a valid and common tool in characterizing of the extremal graph of graph energy.(Quasi-order, i.e., if we compare the correspondingly coefficients of the characteristic polynomial about two given graphs and the signs of these comparisons satisfy coher-ence, then we name that the two graphs meet quasi-order.) However, with the deepening of the study on graph energy, plenty of problems have appeared in extremal energy in which quasi-order method is invalid. Thus, they are called quasi-order incomparable problems. Utilizing Coulson integral formula sufficiently and combining some knowl-edge in analysis, algebra and combinatorics, we successfully solve a series of these problems.Let G be a connected graph with n vertices and m edges. We define the cyclomatic number of a graph G as c(G)=m-n+1. According to this definition, if G meets c(G)=0,1,2or3, then the graph G is called acyclic (or a tree), unicyclic, bicyclic, or tricyclic, respectively.In2001, Gutman and Vidovic proposed the conjecture:if n≥16and n=14, the graph with the maximal energy is Pn6,6in bicyclic graphs. Pn6,6is the graph obtained from two cycles C6joined by a path Pn-12.Denote by Rn-t,t the graph obtained from two cycles Cn-t and Ct (n-t,t≥10and n-t=t=2(mod4)) connected by an edge. Li and Zhang in2007, confirmed the problem in bipartite bicyclic graphs, however, they can’t exclude the graph Rn-t,t·Because the two graphs are quasi-order incomparable. In Subsection2.1.1, we completely solve this problem, i.e., graph Pn6,6has the maximal energy in bipartite bicyclic graphs. In Subsection2.1.2, we further study this problem about the extremal energy for general bicyclic graphs, and verify the above conclusion in the class of graphs ln.In2008, Liu and Zhou characterized the minimal energy graph in l(n). But they can’t determine that which graph has the minimal energy in two quasi-order incompa-rable graphs. In Subsection2.2.1, we settle this puzzle.In2008, Wei considered the problem of the extremal graph of energy in the class of ln. But the author discovered the fact that the graph having the second-minimal energy is one of the two quasi-order incomparable graphs in the energy ordering. In Subsection2.2.2, We solve this problem, and obtain the extremal energy graphs from the minimal graph to seventh-minimal graph. The proofs of the above problems will be exhibited in Chapter2.In1999, Caporossi et al. proposed a conjecture on minimal energy. They also confirmed it for cases m=n-1and m=2(n-1). In2001, Hou showed the conjecture is true for m=n. In2005, Li and Zhang investigated this open problem for m=n+1with a restriction.In2008, Li et al. discussed this open problem in bipartite tricyclic graphs. Unfortunately, they can’t exclude some graphs which are quasi-order incomparable. In Chapter3, we successfully conquer the puzzle.In Chapter4, we will introduce the graph with the minimal energy in a class of unicyclic graphs. |
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