It is well-known that the second smallest eigenvalue 22 of the difference Laplacian matrix of a graph G is related to the expansion properties of G. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance in G in terms of 22 are derived.

In this paper, we study the impact of the introduction of edge weights on the typical distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the (edge) diameter of sparse random graphs on n vertices (under some standard regularity conditions) when the edge weights are i.i.d. exponential random variables of rate one.

We consider an open shop scheduling problem. At first, a bi-objective possibilistic mixed-integer programming formulation is developed. The inherent uncertainty in processing times and due dates as fuzzy parameters, machine-dependent setup times and removal times are the special features of this model. The considered bi-objectives are to minimize the weighted mean tardiness and weighted mean co...