On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs

Abstract

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph G, we construct a generalized power of G, denoted by G k,s , which is obtained from G by blowing up each vertex into a k-set and each edge into a (k − 2s)-set, where2 is non-odd-bipartite if and only if G is non-bipartite, and find that G k, k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph G hold for G k, k 2 . In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs G k, k 2 of fixed order, and prove that 2 + √ 5 is the smallest limit point of the nonodd-bipartite hypergraphs G k, k 2 . In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

Equitable partition theorem of tensors and spectrum of generalized power hypergraphs

“…where e u = uv 1 . By Equalities (8), (9) and the positive vector x, we have λ(B k2,s ) > d v1 ≥ 1 and…”

“…Definition 2.5 [9] Let G = (V, E) be a simple graph. For k ≥ 2 and 1 ≤ s ≤ k 2 , the generalized power graph of G, denoted by G k,s = (V k,s , E k,s ), is the k-uniform hypergraph with the following vertex set and edge set…”

“…If s = k 2 , then G k, k 2 is odd-bipartite if and only if G is bipartite (see [9]). the spectrum of generalized power hypergraphs have been intensively studied (for example, see [7,9,16,17,20]). Recently, Yuan, Qi and Shao [20] proved the following results, which confirms a conjecture of Hu, Qi and Shao [7].…”

Ordering of some uniform supertrees with larger spectral radii

“…The first eight supertrees with the largest spectral radii in T(n, k)Now we are ready to prove the main result Theorem 4.10 which determines that the first eight supertrees with the largest spectral radii in T(n, k) are: T k n ,1 , T k n ,2 , T k n ,3 ,T (1, 1, m − 3), T k n ,4 , T k n ,5 , T k n ,6 and T (1, 2, m − 4).…”

“…They also generalized some basic spectral results from graphs to hypergraphs. The (adjacency) spectrum of uniform hypergraphs were further studied in [9,4,8,17,15].…”