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Lov\'asz and his coauthors defined the notion of microcanonical ground stateenergy $\hat{\mathcal{E}}_\mathbb{a} (G,J)$ -- borrowed from the statisticalphysics -- for weighted graphs $G$, where $\mathbb{a}$ is a probabilitydistribution on $\{1,...,q\}$ and $J$ is a symmetric $q \times q$ matrix withreal entries. We define a new version of the ground state energy,$\hat{\mathcal{E}}^c (G,J)=\inf_{\mathbb{a}\in A_c}\hat{\mathcal{E}}_\mathbb{a}(G,J)$, called lower threshold ground state energy, where $A_c = \{\mathbb{a}:\, a_i\ge c,\,i=1,\dots, q \}$. Both types of energies can be extended forgraphons $W$, the limit objects of convergent sequences of simple graphs. Inthe main result of the paper it is stated that if $0\leq c_1<c_2 \leq 1$, thenthe convergence of the sequences $(\hat{\mathcal{E}}^{c_2/q} (G_n,J))$ for each$J$ implies convergence of the sequences $(\hat{\mathcal{E}}^{c_1/q} (G_n,J))$for each $J$. As a byproduct one can derive in a natural way the testability ofminimum balanced multiway cut densities, that is one of the fundamentalproblems of cluster analysis.

Lower threshold ground state energy and testability of minimal balanced cut density