The Matching Energy Of Bicyclic Graphs

The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. A bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. The thesis consists of two parts of work on characterizing extremal matching energy of bicyclic graphs.1) The graphs with maximum matching energy among all bicyclic graphs with given order are characterized. The schedule is as follows. First we partition the set of bicyclic graphs with given order into two classes according to their structure. Then we find out the graphs with maximum matching energy in each class, where much more elaborate ordering on graphs in each class in terms of matching energy is presented. At last, we compare the two extremal graphs from the classes and come to the the graph attaining maximum matching energy.2) The graph with minimum matching energy among all bicyclic graphs with given order and girth is characterized. We take similar route to our goal. The set of all bicyclic graphs with given order and girth is partitioned into the same two classes as before. Then the respective extremal graphs in each class is given, where some elegant results are presented, such as transformations on graphs decreasing matching energy. After comparing the two minimal graphs in the two classes, we get the graph with minimum matching energy.In summary, we set up the quasi-order of comparing the number of matchings in graphs and then find out the graphs with maximum or minimum matching energy in given class of graphs.