The largest H-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs

详细信息    查看全文

• 作者：Yi-Zheng Fan^a ; ^{fanyz@ahu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Murad-ul-Islam Khan^a ; ^{muradulislam@foxmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ying-Ying Tan^a ; ^b ; ^{tansusan1@ahjzu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C65 ; 15A18
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：504
• 期：Complete
• 页码：487-502
• 全文大小：418 K

文摘

Let G   be a simple graph or hypergraph, and let A(G)$\mathcal{A}(G)$, L(G)$\mathcal{L}(G)$, Q(G)$\mathcal{Q}(G)$ be the adjacency, Laplacian and signless Laplacian tensors of G respectively. The largest H  -eigenvalues (respectively, the spectral radii) of L(G)$\mathcal{L}(G)$, Q(G)$\mathcal{Q}(G)$ are denoted respectively by ${\lambda }_{\max }^{\mathcal{L}}(G)$, ${\lambda }_{\max }^{\mathcal{Q}}(G)$ (respectively, ρ^L(G)${\rho }^{\mathcal{L}}(G)$, ρ^Q(G)${\rho }^{\mathcal{Q}}(G)$). It is known that for a connected non-bipartite simple graph G  , ${\lambda }_{\max }^{\mathcal{L}}(G)={\rho }^{\mathcal{L}}(G)<{\rho }^{\mathcal{Q}}(G)$. But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs $G^{k,\frac{k}{2}}$, which are constructed from simple connected graphs G by blowing up each vertex of G   into a $\frac{k}{2}$-set and preserving the adjacency of vertices.

Suppose that G   is non-bipartite, or equivalently $G^{k,\frac{k}{2}}$ is non-odd-bipartite. We get the following spectral properties: (1) 1f9b2f123e17cdb84903d997ee3">${\rho }^{\mathcal{L}}(G^{k,\frac{k}{2}})={\rho }^{\mathcal{Q}}(G^{k,\frac{k}{2}})$ if and only if k   is a multiple of 4; in this case ${\lambda }_{\max }^{\mathcal{L}}(G^{k,\frac{k}{2}})<{\rho }^{\mathcal{L}}(G^{k,\frac{k}{2}})$. (2) If $k\equiv 2(\mathrm{mod}4)$, then for sufficiently large k  , ${\lambda }_{\max }^{\mathcal{L}}(G^{k,\frac{k}{2}})<{\rho }^{\mathcal{L}}(G^{k,\frac{k}{2}})$. Motivated by the study of hypergraphs $G^{k,\frac{k}{2}}$, for a connected non-odd-bipartite hypergraph G  , we give a characterization of L(G)$\mathcal{L}(G)$ and Q(G)$\mathcal{Q}(G)$ having the same spectra or the spectrum of A(G)$\mathcal{A}(G)$ being symmetric with respect to the origin, that is, L(G)$\mathcal{L}(G)$ and Q(G)$\mathcal{Q}(G)$, or A(G)$\mathcal{A}(G)$ and −A(G)$-\mathcal{A}(G)$ are similar via a complex (necessarily non-real) diagonal matrix with modular-1 diagonal entries. So we give an answer to a question raised by Shao et al., that is, for a non-odd-bipartite hypergraph G  , that L(G)$\mathcal{L}(G)$ and Q(G)$\mathcal{Q}(G)$ have the same spectra can not imply they have the same H-spectra.