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

Linear Algebra and its Applications, 2016, 504: 487-502 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) $\rho^L(G^{k,{k \over 2}}) =\rho^Q(G^{k,{k \over 2}})$ if and only if $k$ is a multiple of $4$; in this case $\lambda_{\max}^L(G^{k,\frac{k}{2}})