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

Abstract

Let $G$ be a simple graph or hypergraph, and let $A(G),L(G),Q(G)$ be the adjacency, Laplacian and signless Laplacian tensors of $G$ respectively. The largest $H$-eigenvalues (resp., the spectral radii) of $L(G),Q(G)$ are denoted respectively by ${\lambda }_{\max }^L(G),{\lambda }_{\max }^Q(G)$ (resp., ${\rho }^L(G),{\rho }^Q(G)$). For a connected non-bipartite simple graph $G$, ${\lambda }_{\max }^L(G)={\rho }^L(G)<{\rho }^Q(G)$. 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) ${\rho }^L(G^{k,\frac{k}{2}})={\rho }^Q(G^{k,\frac{k}{2}})$ if and only if $k$ is a multiple of $4$; in this case ${\lambda }_{\max }^L(G^{k,\frac{k}{2}})<{\rho }^L(G^{k,\frac{k}{2}})$. (2) If $k\equiv 2(\mathrm{mod}4)$, then for sufficiently large $k$, ${\lambda }_{\max }^L(G^{k,\frac{k}{2}})<{\rho }^L(G^{k,\frac{k}{2}})$. Motivated by the study of hypergraphs $G^{k,\frac{k}{2}}$, for a connected non-odd-bipartite hypergraph $G$, we give a characterization of $L(G)$ and $Q(G)$ having the same spectra or the spectrum of $A(G)$ being symmetric with respect to the origin, that is, $L(G)$ and $Q(G)$, or $A(G)$ and $-A(G)$ 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 $L(G)$ and $Q(G)$ have the same spectra can not imply they have the same $H$-spectra.

References in corpus (1)

• The $H$-spectrum of a generalized power hypergraph

Cited by in corpus (5)

• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue
• The spectral property of hypergraph coverings
• The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices
• The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs