On the $α$-spectral radius of uniform hypergraphs

Abstract

For 0 ≤ α < 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.

On some properties of the α-spectral radius of the k-uniform hypergraph

“…The conjecture is already confirmed by Guo in [12]. Furthermore, Guo in [4] showed that Conjecture 1 also holds for connected k-uniform hypergraph when s = 0. Question 1.…”

“…The 2-switch operation is a useful tool for study in graph theory, especially in terms of degree sequence. In [4,16], the perturbation of the adjacency spectral radius and α-spectral radius of a hypergraph under 2-switch operation is studied. Definition 4.1 (2-switching operation, see [16]).…”

“…In [13], Li and Shao showed that for the adjacency spectral radius, the signless Laplacian spectral radius and the incidence Q-spectral radius, S k (1, m − 2) attains uniquely the second largest spectral radius among all k-uniform supertrees on n vertices and m edges. In [4], Guo and Zhou determined the hypergraph with the largest α-spectral radius among all k-uniform supertrees on n vertices. Theorem 6.1.…”

“…In Section 2, notations and some definitions about tensors and hypergraphs are given. In Section 3, applying the results about the perturbation of spectral radii of the hypergraph under moving edges and 2-switching operations, we present a generalization of the result related to Conjecture 1 reported in [4]. In Section 4, we determine the k-uniform non-caterpillar supertrees on n vertices in N C(m, d) with the first two largest A α spectral radii in N C(m, d) and N C(m).…”

“…In 2017, the concept of A α matrix of a graph G was put forward by Nikiforov [1] which is hybrid of A(G) and D(G) similar to the signless Laplacian matrix, where A(G), D(G) are the adjacency matrix and degree matrix of graph G, respectively. The study of A α matrix has started attracting attention of researchers recently (see [2][3][4]).…”