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

Abstract

SUMMARYIn 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 an mth‐order N‐dimensional tensor , we show that the normalized eigenspace associated with maximum Z‐eigenvalue function is ρth‐order Hölder stable at with . As a by‐product, we also establish that the maximum Z‐eigenvalue function is always at least ρth‐order semismooth at . 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/second‐largest Z‐eigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula. Copyright © 2013 John Wiley & Sons, Ltd.