Research On Some Extremal Problems Of Two Types Of Graphs

This paper which stands on the basis of previous results, do further research on sharp bounds of Sums of powers of the degrees of graphs with a given number of vertices and cut edges, also do further research on the Laplace spectral radii of tricyclic graphs.In the first two sections, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discus-sion on the status quo, it fully shows the main work's necessity and innovation.In Section 3, we study an extremal problem on the degree sequences of graphs: determine the minimum and maximum values of Sums of powers of the de-grees of graphs with a given number of n vertices and k cut edges. And the corresponding graphs are characterized.In Section 4, we determines the unique graph with the maximal Laplacian spectral radius among all tricyclic graphs with n vertices and girth g which contain exactly three (resp. four) cycles. Furthermore, when g is even, we can also obtain the upper bound of the Laplacian spectral radius and the extremal graph among all tricyclic graphs with n vertices and girth g.In Section 5, We determine the first four largest Laplacian spectral radii of tri-cyclic graphs with n vertices together with the corresponding extremal graphs.