Ordering Trees With A Given Number Of Non-pendant Edges With Respect To Their Energies

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. To find the extremal energies of a class of graphs or to give an order of its elements with respect to energy is one of the most important research topics in Chemical Graph Theory. We see from the definition of energy that considering the spectrum of a graph is a way of studying energy. On the other hand, for acyclic graphs, the energy of a graph can be expressed as a monotonically increasing function of matching numbers of the graph, which provides us another way of studying energy, specially ordering graphs with respect to energy. Our main results are as follows:1. A class of graphs with special structure are considered. Some order relations with respect to their matching numbers are given. These order relations are the generalization and improvement of some well-known results. Moreover,the maximal and minimal elements in the class of graphs are characterized.2. For the class of trees with n vertices and q non-pendent edges, the structure of trees with the minimal and the second minimal energies are determined. The method is mainly based on two transformations on graphs, which keep the number of non-pendent edges unchanged but decrease the energy of a graph. Moreover, the tree with the maximal energy are considered and, as another proof of the straightforwardness of our method, the order relations of some small graphs are shown.