The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Abstract

Let A be a weakly irreducible nonnegative tensor with spectral radius ρ(A). Let D (respectively, D (0) ) be the set of normalized diagonal matrices arising from the eigenvectors of A corresponding to the eigenvalues with modulus ρ(A) (respectively, the eigenvalue ρ(A)). It is shown that D is an abelian group containing D (0) as a subgroup, which acts transitively on theThe spectral symmetry of A is characterized by the group D/D (0) , and A is called spectral ℓ-symmetric. We obtain the structural information of A by analyzing the property of D, especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover A is symmetric, we prove that A is spectral ℓ-symmetric if and only if it is (m, ℓ)-colorable. We characterize the spectral ℓ-symmetry of a tensor by using its generalized traces, and show that for an arbitrarily given integer m ≥ 3 and each positive integer ℓ with ℓ | m, there always exists an m-uniform hypergraph G such that G is spectral ℓ-symmetric.2000 Mathematics Subject Classification. Primary 15A18, 05C65; Secondary 13P15, 05C15.

Stably computing the multiplicity of known roots given leading coefficients

“…Now, consider the Rowling hypergraph A drawing of is given in Figure where the edges are drawn as arcs and its spectrum is drawn similarly to that of ; note that is also the Fano plane minus two edges. We have It is easy to verify that is not a 3‐cylinder; however, its spectrum, like that of 3‐cylinders, is invariant under multiplication by any third root of unity (see Lemma 3.11 in the work of Fan et al). By the work of Lu et al, we have …”

“…It is easy to verify that  is not a 3-cylinder; however, its spectrum, like that of 3-cylinders, 9 is invariant under multiplication by any third root of unity (see Lemma 3.11 in the work of Fan et al 19 ). By the work of Lu et al, 15 we have…”

Eigenvariety of nonnegative symmetric weakly irreducible tensors associated with spectral radius and its application to hypergraphs

“…But, in general for A being a nonnegative irreducible or weakly irreducible tensor, including the Perron vector, A may have more than one eigenvector corresponding to ρ(A) up to a scalar, i.e. V ρ(A) may contains more than one element [7]. So,it is a natural problem to characterize the dimension or the cardinality of V ρ(A) .…”

“…Under the actions of D and D (0) , Fan et.al [7] get some structural properties of nonnegative weakly irreducible tensors similar to those of nonnegative irreducible matrices. The cyclic index c(A) of a nonnegative weakly irreducible tensor A was given explicitly in [7] by using the generalized traces. In particular, if A is a symmetric tensor of order m, then c(A)|m.…”

Eigenvectors of Laplacian or signless Laplacian of hypergraphs associated with zero eigenvalue

“…Nikiforov [22] applied the symmetric spectrum of the adjacency tensor to characterize the odd-colorable property of a hypergraph. Fan et al [9] characterized the spectral symmetry of the adjacency tensor of a hypergraph by means of the generalized traces which are related to the structure of the hypergraph.…”