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Register a new userPrincipal eigenvector and spectral radius of uniform hypergraphs
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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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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 $Delta+mu$.
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.
The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the $p$-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the $p$-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the $p$-spectral radius model.
The principal ratio of a connected graph $G$, $gamma(G)$, is the ratio between the largest and smallest coordinates of the principal eigenvector of the adjacency matrix of $G$. Over all connected graphs on $n$ vertices, $gamma(G)$ ranges from $1$ to $n^{cn}$. Moreover, $gamma(G)=1$ if and only if $G$ is regular. This indicates that $gamma(G)$ can be viewed as an irregularity measure of $G$, as first suggested by Tait and Tobin (El. J. Lin. Alg. 2018). We are interested in how stable this measure is. In particular, we ask how $gamma$ changes when there is a small modification to a regular graph $G$. We show that this ratio is polynomially bounded if we remove an edge belonging to a cycle of bounded length in $G$, while the ratio can jump from $1$ to exponential if we join a pair of vertices at distance $2$. We study the connection between the spectral gap of a regular graph and the stability of its principal ratio. A naive bound shows that given a constant multiplicative spectral gap and bounded degree, the ratio remains polynomially bounded if we add or delete an edge. Using results from matrix perturbation theory, we show that given an additive spectral gap greater than $(2+epsilon)sqrt{n}$, the ratio stays bounded after adding or deleting an edge.
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin-Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin-Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in [19] and a result in [11].
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