The Adjacency and Signless Laplacian Spectra of Cored Hypergraphs and Power Hypergraphs
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The Adjacency and Signless Laplacian Spectra of Cored Hypergraphs and Power Hypergraphs Jun-Jie Yue1,2 · Li-Ping Zhang1 · Mei Lu1 · Li-Qun Qi3
Received: 25 May 2016 / Revised: 26 August 2016 / Accepted: 31 October 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs. We investigate the properties of their adjacency and signless Laplacian H-eigenvalues. Especially, we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids. We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra. Keywords H-eigenvalue · Hypergraph · Adjacency tensor · Signless Laplacian tensor · Sunflower · Squid Mathematics Subject Classification 74B99 · 15A18 · 15A69
This work was supported by the National Natural Science Foundation of China (No. 11271221) and the Specialized Research Fund for State Key Laboratories.
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Li-Ping Zhang [email protected] Jun-Jie Yue [email protected] Mei Lu [email protected] Li-Qun Qi [email protected]
1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2
State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100910, China
3
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
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J.-J. Yue et al.
1 Introduction In recent years, the study of spectral hypergraph theory via tensors [1–8] has attracted extensive attention and interest since the work of [1,4,8,9]. As was in [9], a real tensor T = (ti1 ,··· ,ik ) of order k and dimension n refers to a multidimensional array (also called hypermatrix) with entries ti1 ,··· ,ik such that ti1 ,··· ,ik ∈ R n k−1 is defined for all i j ∈ [n] := {1, · · · , n} and j ∈ [k]. Given a vector x ∈ R , T x as an n-dimensional vector such that its ith element is i2 ,··· ,ik ∈[n] tii2 ···ik xi2 · · · xik for i ∈ [n]. Let I be the identity tensor of appropriate dimension, e.g., i i1 ···ik = 1 if and only if i 1 = · · · = i k ∈ [n], and zero otherwise when the dimension is n. The following definition was introduced by Qi [9]. Definition 1.1 Let T be a k-th order n-dimensional real tensor. For some λ ∈ R, if polynomial system (λI − T )x k−1 = 0 has a solution x ∈ Rn \{0}, then λ is called an H-eigenvalue and x an H-eigenvector. Obviously, H-eigenvalues are real number. By [9,10], we have the number of Heigenvalues of a real tensor as finite. By [8], we have that all the tensors considered in this paper have at least one H-eigenvalue. Hence, we can denote λ(T ) as the largest H-eigenvalue of a real tensor T . As was in [8], a hypergraph means an undirected simple k-uniform hypergraph G with vertex set V, which is labeled as [n], and edge set E. By k-uniformity, we mean that for every edge e ∈ E, the cardinality |e| of e is equal to k. Throughout this paper, k 3
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