一般超图规范化拉普拉斯张量的谱性质

张磊; 任海珍

doi:10.13451/j.sxu.ns.2024128

山西大学学报(自然科学版) ›› 2025, Vol. 48 ›› Issue (04) : 705 -712. DOI: 10.13451/j.sxu.ns.2024128

基础数学与应用数学

一般超图规范化拉普拉斯张量的谱性质

张磊 ¹^,²^,³ ,
任海珍 ¹^,²

作者信息 +

The Properties of the Spectra of Normalized Laplace Tensor in General Hypergraphs

Lei ZHANG ¹^,²^,³ ,
Haizhen REN ¹^,²

Author information +

文章历史 +

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摘要

超图的规范化拉普拉斯张量在涉及超图模型的多类聚类问题中有着广泛的应用。本文证明了如果H是rank(H)=m的连通一般超图，则H的邻接张量谱是对称的，当且仅当m是偶数且H是奇着色的。在此基础上，利用一般超图的奇着色性，给出了一般超图的规范化拉普拉斯张量谱半径为2的刻画。

Abstract

The normalized Laplacian tensor of hypergraphs has a wide range of applications in multi class clustering problems involving hypergraph models. In this paper, we prove that if $H$ is a connected general hypergraph with rank ( $H$ )= $m$ , then the spectrum of the adjacency tensor of $H$ is symmetric if and only if $m$ is even and $H$ is odd-colorable. Furthermore, we give a characterization of the general hypergraphs with the normalized Laplace spectral radius 2 in terms of the odd-colorability of general hypergraphs.