Minimal extremal graphs for addition of algebraic connectivity and independence number of connected graphs

Let G = (V, E) be a simple connected graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of graph G is L(G) = D(G) − A(G). Let a(G) and α(G), respectively, be the second smallest Laplacian eigenvalue and the independence number of graph G. In this paper, we characterize the extremal graph with second minimum value for addition of algebraic connectivity and independence number among all connected graphs with n ≥ 6 vertices (Actually, we can determine the p-th minimum value of a(G)+α(G) under certain condition when p is small). Moreover, we present a lower bound to the addition of algebraic connectivity and radius of connected graphs.