A bound on the spectral radius of hypergraphs with e edges

Abstract

For r ≥ 3, let fr : [0, ∞) → [1, ∞) be the unique analytic function such that fr( k r ) = k−1 r−1 for any k ≥ r − 1. We prove that the spectral radius of an r-uniform hypergraph H with e edges is at most fr(e). The equality holds if and only if e = k r for some positive integer k and H is the union of a complete r-uniform hypergraph K r k and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs. √ 1+8e−1 2. The equality holds if and only if e = k 2 and G is the union of the complete graph K k and some isolated vertices. Friedland [7] proved a bound which is tight on the complete graph with one, two, or three edges removed or the complete graph with one edge added. Rowlinson [17] finally confirmed Brualdi and Hoffman's conjecture, and proved that G e attains the maximum spectral radius among all graphs with e edges.

Spectral Extremal Results for Hypergraphs

“…In 2005, Qi [25] and Lim [20] independently introduced the concept of tensor eigenvalues and the spectra of tensors. An order k dimension n real tensor 1…”

“…Much of recent work in spectral extremal graph theory is due to Nikiforov, who has considered maximizing the spectral radius over several families of graphs; see [23]. Although the area is still difficult and underdeveloped, we believe that ultimately a spectral approach to extremal hypergraph theory will turn out to be a fruitful and interesting accompaniment to "conventional" extremal theory, see e.g., [1,17].…”

Spectral radius of {0,1}-tensor with prescribed number of ones

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On the Effect of Data Dimensionality on Eigenvector Centrality

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