  In this work we present the notion of greyscale of a graph as a colouring of
its vertices that uses colours from the real interval [0,1]. Any greyscale
induces another colouring by assigning to each edge the non-negative difference
between the colours of its vertices. These edge colours are ordered in
lexicographical decreasing ordering and gives rise to a new element of the
graph: the gradation vector. We introduce the notion of minimum gradation
vector as a new invariant for the graph and give polynomial algorithms to
obtain it. These algorithms also output all greyscales that produce the minimum
gradation vector. This way we tackle and solve a novel vectorial optimization
problem in graphs that may produce more satisfactory solutions than those ones
generated by known scalar optimization approaches.
  The interest of these new concepts lies in their possible applications for
solving problems of engineering, physics and applied mathematics which are
modeled according to a network whose nodes have assigned numerical values of a
certain parameter delimited by a range of real numbers. The objective is to
minimize the differences between each node and its neighbors, ensuring that the
extreme values of the interval are assigned.
