Research On Some Extremal Problems Of Some Classes Of Graphs

This paper which stands on the basis of previous results, do further research on some extremal problems of several classes of graphs.In the first two sections, we first introduce the background and significance of the research, including the development regarding this aspect, which fully shows the main work's necessity and innovation. Then, we give some basic conceptions, notations and lemmas which can be used in our paper.In Section 3 (resp.Section 4), we study the sharp bounds on function f(G,α) of conjugated unicyclic (resp.bicyclic) graphs. And characterize the correspond-ing graphs. In Section 5, sharp bounds on function f(G,2) and h(G,1) of bipartite graphs with a given diameter are determined. And the corresponding graphs are characterized. Then, we also determine the largest, second largest and small-est value on function f(G,2) and h(G,1) of all bipartite graphs. And the corresponding graphs are characterized. In Section 6, we introduce the signless Laplacian index of cacti with k-pendent vertices, determine its upper bound and the corresponding graphs. Then, the upper bounds on the signless Laplacian index of cacti and cacti with perfect matching are also determined. And the corresponding graphs are character-ized.