The largest α-sepctral radius of k-uniform bicyclic hypergraphs

Let G be a k-uniform hypergraph and A? (G) = ?D(G)+(1-?)A(G) the convex linear combination of its degree diagonal tensor D(G) and its adjacency tensor A(G), where k ? 3 and 0 ? ? < 1. The ?-spectral radius of G is the largest modulus of all the eigenvalues of A? (G). Let B(n, k) be the set of the connected k-uniform bicyclic hypergraphs, where k ? 3. The number of the edges of the hypergraphs in B(n, k) is denoted by m=n+1/k-1. We develop a new ??-normal labeling method for calculating the ?-spectral radius of k-uniform hypergraphs. By using some transformations and the new ??-normal labeling methods, we characterize the hypergraphs with the first and the second largest ?-spectral radii among B(n, k), where k ? 4 and m = n+1/k?1 ? 20.