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Register a new userThe bounds of the spectral radius of general hypergraphs in terms of clique number
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The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.
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For $0leq alpha < 1$, the $mathcal{A}_{alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $mathcal{A}_{alpha}(G):=alpha mathcal{D}(G)+(1-alpha) mathcal{A}(G)$, where $mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $mathcal{A}_{alpha}$-spectral radius and an average degree $frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the $mathcal{A}_{alpha}$-spectral radius are obtained in terms of the maximum and minimum degrees of a hypergraph.
Let $G$ be a connected uniform hypergraphs with maximum degree $Delta$, spectral radius $lambda$ and minimum H-eigenvalue $mu$. In this paper, we give some lower bounds for $Delta-lambda$, which extend the result of [S.M. Cioabu{a}, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory, Ser. B 97 (2007) 483-486] to hypergraphs. Applying these bounds, we also obtain a lower bound for $Delta+mu$.
There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin and Straus result to hypergraphs is false. Frankl and F{u}redi conjectured that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges. For $r=2$, Motzkin and Straus theorem confirms this conjecture. For $r=3$, it is shown by Talbot that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $3$-uniform hypergraphs. As an application of this connection, we confirm that Frankl and F{u}redis conjecture holds for bigger ranges of $m$ when $r$=3. We also obtain two weak
In this paper, we investigate spectral properties of the adjacency tensor, Laplacian tensor and signless Laplacian tensor of general hypergraphs (including uniform and non-uniform hypergraphs). We obtain some bounds for the spectral radius of general hypergraphs in terms of vertex degrees, and give spectral characterizations of odd-bipartite hypergraphs.
In this paper, we give some bounds for principal eigenvector and spectral radius of connected uniform hypergraphs in terms of vertex degrees, the diameter, and the number of vertices and edges.
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