THE SHARP LOWER BOUND FOR THE SPECTRAL RADIUS OF CONNECTED GRAPHS WITH THE INDEPENDENCE NUMBER

Abstract

In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize all extremal connected graphs which attain the minimum value among the spectral radii of all connected graphs with order $n=k\alpha$ and the independence number $\alpha$. Moreover, all extremal graphs which attain the maximum value among the spectral radii of clique trees with order $n=k\alpha$ and the independence number $\alpha$ are characterized.