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Title: | Hook immanantal inequalities for Laplacians of trees |

Authors: | Chan, O. Lam, T.K. |

Issue Date: | Aug-1997 |

Citation: | Chan, O.,Lam, T.K. (1997-08). Hook immanantal inequalities for Laplacians of trees. Linear Algebra and Its Applications 261 (1-3) : 23-47. ScholarBank@NUS Repository. |

Abstract: | For an irreducible character χλ of the symmetric group Sn, indexed by the partition λ, the immanant function dλ, acting on an n × n matrix A = (aij), is defined as dλ(A) = ∑σ ∈ Sn χλ(σ)Πn i = 1_ aiσ(i). The associated normalized immanant d̄λ is defined as d̄λ = dλ/χλ(identity) where identity is the identity permutation. P. Heyfron has shown that for the partitions (k, 1n - k), the normalized immanant d̄k satisfies det A = d̄1(A) ≤ d̄2(A) ≤ ⋯ ≤ d̄n(A) = per A (1) for all positive semidefinite Hermitian matrices A. When A is restricted to the Laplacian matrices of graphs, improvements on the inequalities above may be expected. Indeed, in a recent survey paper, R. Merris conjectured that d̄n - 1(A) ≤ n - 2/n - 1 d̄n(A) (2) whenever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of inequalities in (1) when A is the Laplacian matrix of a tree, that includes (2) as a special case. These inequalities are sharp and equality holds if and only if A is the Laplacian matrix of the star. This is proved via the inequalities d̄k(A) - d̄k - 1(A) ≤ d̄k + 1(A) - d̄k(A) for k = 2,3, . . ., n - 1, where A is the Laplacian matrix of a tree. © Elsevier Science Inc., 1997. |

Source Title: | Linear Algebra and Its Applications |

URI: | http://scholarbank.nus.edu.sg/handle/10635/103385 |

ISSN: | 00243795 |

Appears in Collections: | Staff Publications |

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