Ordering Of Graphs With A Perfect Matching By Their Energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. The investigation on the graphs with extremal energies is of theoretical interest and practical importance in the subject of chemical graph theory. The larger the value of enery, the greater the thermodynamic stability of the corresponding compund.Conjugated molecules in chemistry may be classified into two groups: Kekulean and non-Kekulean molecules, depending on whether or not they possess Kekulean structures, i.e., the perfect matchings in graph theory. The graphs with perfect matchings possess many chemical properties. For example, the existence of perfect matchings is closely connected to the stability of aromatic systems.In view of the significance of the graphs with perfect mathings in chemical graph theory, this kind of the graphs with extremal energies is investigated in this thesis. The vertex number of the graphs considered is denoted by In. The main results can be divided into five parts as follows.Firstly, the ordering of the minimal energies of the trees with a perfect matching having degrees no greater than three is considered. By means of a simpler method than that of Li (J. Math. Chem. 25 (1999) 145-169), we obtain the first 2n - 2r - 5 trees in the increasing order of their energies within the class under consideration for n + 1≥14, where r is determined by n + 1≡r (mod 4). The number of the trees obtained here exceeds the reported result (Li 1999) by n - r - 6. We also get a lot of preceding trees in the increasing order of their energies within the class for 6≤n + 1≤13.Secondly, the extremal energies of the trees with a perfect matching having a given diameter d are obtained. The graphs with minimal energies are given for 4≤d≤10. As d = 5, we obtain the last (1 +(?))/2 and y(?)+1 trees in the increasing order of their energies within the class under consideration for n = 2h and n = 2h + 1, respectively, where h≥2.Thirdly, the ordering of the minimal energies of the trees with a perfect matching is studied. We employ a simpler method than that of Zhang & Li (Discrete Appl. Math. 92 (1999) 71-84) to find the trees having the minimal, the second-minimal, and the third-minimal energies.Next, the ordering of unicyclic graphs with perfect matchings having degrees no greater than three by their Hosoya indexes is considered. Four special cases in the increasing order of their Hosoya indexes are studied. The preceding graphs in the increasing order of their Hosoya indexes are determined. Furthermore, the corresponding graphs with minimal energies are provided.Finally, unicyclic graphs with perfect matchings having minimal energy are considered . The corresponding graph with minimal energy is mathematically verified.