Spectral bipartite Turan problems on linear hypergraphs

Abstract

Let $F$ be a graph, and let ${\mathcal{B}}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for ${\mathcal{B}}_r(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of ${\mathcal{B}}_r(K_{2,t})$-free or $\{{\mathcal{B}}_r(K_{s,t}),{\mathcal{B}}_r(C_3)\}$-free linear $r$-uniform hypergraphs, where $C_3$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\{{\mathcal{B}}_r(K_{s,t}),{\mathcal{B}}_r(C_3)\}$-free linear $r$-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

References in corpus (4)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions
• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• Eigenvariety of Nonnegative Symmetric Weakly Irreducible Tensors Associated with Spectral Radius and Its Application to Hypergraphs

Cited by in corpus (1)

• Some Turán-type results for the signless Laplacian spectral radius