Minimum values of the second largest Q-eigenvalue

For a graph G, the signless Laplacian matrix Q(G) is defined as Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) the diagonal matrix whose main entries are the degrees of the vertices in G. The Q-spectrum of G is that of Q(G). In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q₂(G) of a connected graph G. We find the five smallest values of q₂(G) over the set of connected graphs G with given order n. We also characterize the corresponding extremal graphs.