Lov\'asz and his coauthors defined the notion of microcanonical ground state
energy E^a(G,J) -- borrowed from the statistical
physics -- for weighted graphs G, where a is a probability
distribution on {1,...,q} and J is a symmetric q×q matrix with
real entries. We define a new version of the ground state energy,
E^c(G,J)=infa∈AcE^a(G,J), called lower threshold ground state energy, where Ac={a:ai≥c,i=1,…,q}. Both types of energies can be extended for
graphons W, the limit objects of convergent sequences of simple graphs. In
the main result of the paper it is stated that if 0≤c1<c2≤1, then
the convergence of the sequences (E^c2/q(Gn,J)) for each
J implies convergence of the sequences (E^c1/q(Gn,J))
for each J. As a byproduct one can derive in a natural way the testability of
minimum balanced multiway cut densities, that is one of the fundamental
problems of cluster analysis.Comment: 14 page