4 research outputs found
On the -spectral radius of hypergraphs
For real and a hypergraph , the -spectral radius
of is the largest eigenvalue of the matrix , where is the adjacency matrix of , which is a
symmetric matrix with zero diagonal such that for distinct vertices of
, the -entry of is exactly the number of edges containing both
and , and is the diagonal matrix of row sums of . We study
the -spectral radius of a hypergraph that is uniform or not necessarily
uniform. We propose some local grafting operations that increase or decrease
the -spectral radius of a hypergraph. We determine the unique
hypergraphs with maximum -spectral radius among -uniform hypertrees,
among -uniform unicyclic hypergraphs, and among -uniform hypergraphs with
fixed number of pendant edges. We also determine the unique hypertrees with
maximum -spectral radius among hypertrees with given number of vertices
and edges, the unique hypertrees with the first three largest (two smallest,
respectively) -spectral radii among hypertrees with given number of
vertices, the unique hypertrees with minimum -spectral radius among the
hypertrees that are not -uniform, the unique hypergraphs with the first two
largest (smallest, respectively) -spectral radii among unicyclic
hypergraphs with given number of vertices, and the unique hypergraphs with
maximum -spectral radius among hypergraphs with fixed number of pendant
edges
On the α-Spectral Radius of Uniform Hypergraphs
For 0 ≤ α ---lt--- 1 and a uniform hypergraph G, the α-spectral radius of G is the largest H-eigenvalue of αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal tensor of degrees and the adjacency tensor of G, respectively. We give upper bounds for the α-spectral radius of a uniform hypergraph, propose some transformations that increase the α-spectral radius, and determine the unique hypergraphs with maximum α-spectral radius in some classes of uniform hypergraphs