The Method of Eigenvalue Interlacing for Graphs is used to investigate some problems on graphs, such as the lower bounds for the spectral radius of graphs. In this paper, two new sharp lower bounds on the spectral radius of graphs are obtained, and a relation between the Laplacian spectral radius of a graph and the number of quadrangles in the graph is deduced.
It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜（q - 1）.
