The Ordering Of Unicyclic Graphs In Terms Of Their Some Invariants

Let c be a simple connected graph with vertices. We denoted by the adjacency matrix of . All of the eigenvalue of are real, since is real and symmetric. We denote the eigenvalues of in decreasing order byand call then the eigenvalues of. The energy ofis defined as .In this thesis we discuss the problem of the ordering of unicyclic graphs. In terms of their largest eigenvalues and energy. this thesis are divided into two charpters: In the first charpter, we investigate the problem of the ordering of unicyclic graphs by their largest eigenvalues. Let be the graph obtained from the graph by adding pendant edges to one of its vertices. Hong Yuan has proved that in all unicyclic graphs, the largest eigenvalue of the graph is the greatest. On this basis, we discussed the problem further and found the other first five graphs in this order.In the second charpter, we discussed the ordering of the unicyclic graphs by their energy. Let be the graph obtained from cycle by adding pendant edges to one of its vertices, and be the graph obtained from cycle by adding pendant edges and a pendant edge to two vertices of respectively. Yaoping Hou has proved that if is a unicyclic graph with vertices and , then . On this basis, we discussed the problem further and deduced two results as follows:(1) If ,, then , with equality if andonly if 。(2) If ,,then, with equality if and only if...