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

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}})