The previous conditions can be written in term of constrain compatible with ILP as (8.29) If the condition (8.29) is a constrain of the problem, we are sure that only one element is different from zero among the elements of a column and of a row. The last step is to express the fact that the path of an GLSP has to be made by adjacent links, that is, the set of nonzero elements syijkhv with i, j, V fixed has to be constituted by concatenated elements like (Sinch ijhkv This condition can be expressed by the use of the adjacency matrix. From the nodes adjacency matrix, a links adjacency matrix [s]]] can be easily derived, whose elements Bijk are equal to one only if the two adjacent links i, j, and j, k both exist. In particular, it is immediately seen that Thus the condition we are searching is written as (8.30) The meaning of this constraint is the following. Starting from syijikon which are the unique nonzero elements with the first and third index equal to i, the second equal to j and the fifth equal to v, a chain of elements of [sR] can be constructed by adding to each element the following element having the same first, second, and fifth index and the fourth index moved at the third place. These last conditions also define a unique nonzero element, SO that, if a valid path has been found between i and j, it can be constructed in this way. This also implies that both Siinku and spijkxv" representing adjacent links of the path, are equal to 1. From this property derives the fact that Equation 8.26 is satisfied. A similar continuity constraint has to be written across a node for the wavelengths that are not passed to the upper layer by the OXC. In particular, the number of passing through wavelengths entering the OXC has to be equal to the number of passing through wave- lengths exiting the OXC. This condition however is automatically satisfied by the coexistence of the constrain (8.18) and by the property Wk = Whk deriving from the bidirectional nature of the GMPLS GLSPs. The equations we have derived up to now 