Circulant tensors with applications to spectral hypergraph theory and stochastic process

Zhongming Chen, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

17 Citations (Scopus)

Abstract

Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B0 tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is mth order stationary, where m is even, then its mth order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for an even order circulant tensor to be positive semi-definite.
Original languageEnglish
Pages (from-to)1227-1247
Number of pages21
JournalJournal of Industrial and Management Optimization
Volume12
Issue number4
DOIs
Publication statusPublished - 1 Jan 2016

Keywords

  • Circulant hypergraphs
  • Circulant tensors
  • Directed circulant hypergraphs
  • Eigenvalues of tensors
  • Positive semi-definiteness

ASJC Scopus subject areas

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

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