Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs

Abstract

We give sharp upper bounds for the ordinary spectral radius and signless Laplacian spectral radius of a uniform hypergraph in terms of the average 2-degrees or degrees of vertices, respectively, and we also give a lower bound for the ordinary spectral radius. We also compare these bounds with known ones.then ρ is called an eigenvalue of T , and x an eigenvector of T corresponding to ρ, see [7,8]. Let ρ(T ) be the largest modulus of the eigenvalues of T .Let G be a hypergraph with vertex set V (G) = [n] and edge set E(G), see [1]. If every edge of G has cardinality k, then we say that G is a k-uniform hypergraph. Throughout this paper, we consider k-uniform hypergraphs on n vertices with 2 ≤ k ≤ n. A uniform hypergraph is a hypergraph that is k-uniform for some k. For i ∈ [n], E i denotes the set of edges of G containing i. The degree of a vertex i in G is defined asthen G is called a regular hypergraph (of degree d). For i, j ∈ V (G), if there is a sequence of edges e 1 , . . . , e r such that i ∈ e 1 , j ∈ e r and e s ∩ e s+1 = ∅ for all s ∈ [r − 1], then we say that i and j are connected. A hypergraph is connected if every pair of different vertices of G is connected.The adjacency tensor of a k-uniform hypergraph G on n vertices is defined as the tensor A(G) of order k and dimension n whose (i 1 . . . i k )-entry iswhere e 1 = {1, 2, 5}, e 2 = {1, 2, 6}, e 3 = {1, 2, 7}, e 4 = {1, 2, 8}, e 5 = {1, 2, 9}, e 6 = {1, 2, 10}, e 7 = {1, 2, 11}, e 8 = {1, 2, 12}, e 9 = {1, 2, 13}, e 10 = {3, 4, 5}, e 11 = {3, 4, 6}, e 12 = {3, 4, 7}, e 13 = {3, 4, 8}, e 14 = {3, 4, 9}, e

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