A positive square-energy strengthening of Turán's theorem

What the math gives to ML

The paper gives a sharp upper bound on the total squared energy carried by positive adjacency eigenvalues, not merely on the spectral radius. For an n-vertex graph with clique number r, the positive spectral energy obeys \(\sqrt{s^+(A)}\le (1-1/r)n\), while the Motzkin–Straus variational form characterizes the same clique-dependent quantity through a simplex-constrained quadratic maximization. This suggests a graph-neural-network regularizer for learned adjacency or attention graphs: constrain their positive spectral energy to match a desired sparsity/clique-complexity budget, rather than controlling only the largest eigenvalue. The exact theorem is guaranteed for unweighted graphs; applying it to continuously learned nonnegative symmetric matrices is an experimentally testable relaxation.