The largest $\alpha$-sepctral radius of $k$-uniform bicyclic hypergraphs

Abstract

Let $\mathcal G$ be a $k$-uniform hypergraph and   $\bm{\mathcal A}_\alpha(\mathcal G)=\alpha \bm{\mathcal D}(\mathcal G) + (1 -\alpha)\bm{\mathcal A}(\mathcal G)$ the convex linear combination  of its  degree diagonal tensor $\bm{\mathcal D}(\mathcal G)$ and its adjacency tensor $\bm{\mathcal A}(\mathcal G)$, where $k\geq 3$ and $0\leq \alpha<1$.  The $\alpha$-spectral radius of $\mathcal{G}$ is the largest modulus of all the eigenvalues of $ \bm{\mathcal A}_{\alpha}(\mathcal{H})$.   Let $\mathcal B(n,k)$ be the set of the connected $k$-uniform bicyclic hypergraphs, where $k\geq 3$.   The number of the edges of the hypergraphs in $\mathcal B(n,k)$ is denoted by $m=\frac{n+1}{k-1}$.  We develop a new $\rho_{\alpha}$-normal labeling method for calculating  the $\alpha$-spectral radius of $k$-uniform hypergraphs.  By using some transformations  and the new $\rho_{\alpha}$-normal labeling methods, we characterize the hypergraphs with the first and the second  largest $\alpha$-spectral radii among  $\mathcal B(n,k)$, where  $k\geq 4$ and $m=\frac{n+1}{k-1}\geq20$.