Of all trees on given order n,diameter d and pendant vertices number s, which achieves the maximal spectral radius? Let Tn,d,s be a tree with order n,diameter d and pendant vertex number s obtained by spanning s-2 paths of almost equal lengths at r th-vertex of fixed path Pd+1,where r=r(d) is the integer part on(d+1)/2.

The minimum and maximum values of Randic\' index for the chemical trees with given order and number of pendent vertex are obtained by Hansen and Mélot,and the corresponding chemical trees are characterized.

The numbers of physical cover of the cover system in NMM when the crack was extended were amended through the setting of hanging nodes.

The author studies an explicit fully discrete spectral method and pseudo spectral method with weaker stability condition for the Cahn Hilliard equation.

In this thesis we investigate some graphic transformations and properties of extremal graphs on the Randic index among all trees with k pendant vertices, and use some of these transformations to study the problem of extremal values of the Randic index.