The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory

In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Z-eigenvalues of a real symmetric tensor with even order. We first establish that the maximum Z-eigenvalue function is a continuous and convex function on the symmetric tensor space, and so, provide formulas of the convex conjugate function and ε-subdifferential of the maximum Z-eigenvalue function. Consequently, for anmth order n-dimensional tensorA, we show that the normalized eigenspace associated with maximum Z-eigenvalue function is ρth-order Hölder stable at A with ρ = 1 m(3m−3)n−1−1 . As a byproduct, we also establish that the maximum Z-eigenvalue function is always at least ρth-order semismooth at A. As an application, we introduce the characteristic tensor of a hypergraph and show that the maximum Z-eigenvalue function of the associated characteristic tensor provides a natural link for the combinatorial structure and the analytic structure of the underlying hypergraph. Finally, we establish a variational formula for the second largest Z-eigenvalue for the characteristic tensor of a hypergraph and use it to provide lower bounds for the bipartition width of a hypergraph. Some numerical examples are also provided to show how one can compute the largest/2nd-largest Z-eigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula. Copyright c ⃝ 2013 John Wiley & Sons, Ltd.