The maximum spectral radius of the weighted bicyclic hypergraphs (Q6586414)

Let \(G = (V (G),E(G),W(G))\) be a weighted hypergraph, where \(V (G)\) is the vertex set, \(E(G) = \{e_1,e_2,\ldots,e_m\}\) the edge set with \(e_i\subset V(G)\) for \(i = 1,\dots ,m\) and \(W(G) = \{w_G(e) \in\mathbb{R}\mid e \in E(G)\}\) the set of weights of \(G\). Then \(e_i\) with \(1\leq i\leq m\) is called an edge of \(G\). If \(|e_i|=k\) for \(1\leq i\leq m\), then \(G\) is called a weighted \(k\)-uniform hypergraph.\N\NLet \(\mathbb{B}_{n,k,a}\) be the set of the connected weighted \(k\)-uniform bicyclic hypergraphs on \(n\) vertices with positive integer weights and fixed total weight sum \(a\), where \(k\geq 4\) and \(a\geq \frac{n+1}{k-1}\geq 3\). The main result characterizes hypergraphs in \(\mathbb{B}_{n,k,a}\) with maximum spectral radius. The spectral radius of \(G\) is the spectral radius of the adjacency tensor of \(G\).