more inequalities for laplacian indices by way of majorization

The ntuple of laplacian characteristic values of a graph is majorized by the conjugate sequence of its degrees. using that result we find a collection of general inequalities for a number of laplacian indices expressed in terms of the conjugate degrees, and then with a maximality argument, we find tight general bounds expressed in terms of the size of the vertex set n and the average degree dg = 2|e|/n. we also find some particular tight bounds for some classes of graphs in terms of customary graph parameters.