# On the α-spectral radius of hypergraphs

For real α∈ [0,1) and a hypergraph G, the α-spectral radius of G is the largest eigenvalue of the matrix A_α(G)=α D(G)+(1-α)A(G), where A(G) is the adjacency matrix of G, which is a symmetric matrix with zero diagonal such that for distinct vertices u,v of G, the (u,v)-entry of A(G) is exactly the number of edges containing both u and v, and D(G) is the diagonal matrix of row sums of A(G). We study the α-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the α-spectral radius of a hypergraph. We determine the unique hypergraphs with maximum α-spectral radius among k-uniform hypertrees, among k-uniform unicyclic hypergraphs, and among k-uniform hypergraphs with fixed number of pendant edges. We also determine the unique hypertrees with maximum α-spectral radius among hypertrees with given number of vertices and edges, the unique hypertrees with the first three largest (two smallest, respectively) α-spectral radii among hypertrees with given number of vertices, the unique hypertrees with minimum α-spectral radius among the hypertrees that are not 2-uniform, the unique hypergraphs with the first two largest (smallest, respectively) α-spectral radii among unicyclic hypergraphs with given number of vertices, and the unique hypergraphs with maximum α-spectral radius among hypergraphs with fixed number of pendant edges.

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