Exponent-Incidence Constraints for Tensor Eigenvectors of Multi-Hypergraphs

We study support-determined constraints for eigenvectors of nonnegative symmetric tensors whose support may contain repeated indices. Such tensors are naturally encoded by uniform multi-hypergraphs, where each multiedge is represented by an exponent vector \( \alpha\in\mathbb N_0^n\ \) with \( |\alpha|=k\ \). Replacing the ordinary vertex-edge incidence matrix by the exponent-incidence matrix, we show that every nonzero-eigenvalue H-eigenvector satisfies linear incidence constraints in the transformed coordinates \( y_i=x_i^k\ \). These constraints are invariant under positive scalar edge weights and reduce to the usual support-incidence constraints for ordinary squarefree hypergraphs. We also describe the positive branch of the resulting constraint variety and prove a positive-weight realization criterion: a positive vector \( x\ \) can be realized as an eigenvector of some positive edge-weighting of a fixed multi-hypergraph if and only if \( x^{[k]}\ \) lies in the positive cone generated by the exponent-incidence columns. Thus the exponent-incidence constraint variety gives a linear algebraic relaxation, while the positive exponent-incidence cone gives the exact positive-weight realization region.