[Paper Review] The p-spectral radius of k-partite and k-chromatic uniform hypergraphs

Liying Kang, Vladimir Nikiforov+1 more|arXiv (Cornell University)|Jan 31, 2014

graph theory and CDMA systems9 references3 citations

TL;DR

This paper establishes extremal bounds for the $p$-spectral radius of $k$-partite and $k$-chromatic $r$-uniform hypergraphs, proving that the complete $k$-partite hypergraph $T_k^r(n)$ and the complete $k$-chromatic hypergraph $Q_k^r(n)$ maximize the $p$-spectral radius for all $p > 1$ and $p \geq 1$, respectively, under equal vertex partitioning. The results generalize classical extremal graph theory to hypergraphs using spectral methods.

ABSTRACT

We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.

Motivation & Objective

• To determine the maximum $p$-spectral radius among $k$-partite $r$-uniform hypergraphs of order $n$ for $p > 1$.
• To extend known extremal results for $p=1$ and $p=2$ to general $p \geq 1$ in the context of $k$-chromatic hypergraphs.
• To resolve extremal problems for $p$-spectral radius in $k$-partite and $k$-chromatic hypergraphs, generalizing prior work on Lagrangian and spectral radius.
• To establish tight upper bounds on the $p$-spectral radius that are achieved uniquely by the complete $k$-partite and $k$-chromatic hypergraphs.

Proposed method

• Define the $p$-spectral radius $\lambda^{(p)}(G)$ as the maximum of the multilinear form $P_G(\mathbf{x})$ over the $\ell^p$-unit sphere.
• Use the method of Lagrange multipliers and symmetry arguments to show that the maximum is achieved when vertex sets are as equal as possible.
• Apply the Power Mean inequality to bound the $p$-norm of monomial terms in the spectral form.
• Leverage convexity of the function $f(x) = \binom{x}{r}$ for $x > r-1$ to bound the number of edges in $k$-chromatic hypergraphs.
• Prove that equality in the extremal bounds occurs only when the hypergraph is complete $k$-partite or complete $k$-chromatic with balanced parts.
• Use eigenvalue and eigenvector analysis to relate $\lambda^{(1)}(G)$ and $\lambda^{(p)}(G)$, and derive bounds via interpolation and Hölder-type inequalities.

Experimental results

Research questions

• RQ1 What is the maximum $p$-spectral radius of a $k$-partite $r$-uniform hypergraph of order $n$ for $p > 1$?
• RQ2 Does the complete $k$-partite $r$-graph $T_k^r(n)$ uniquely maximize $\lambda^{(p)}(G)$ among $k$-partite $r$-graphs for $p > 1$?
• RQ3 What is the maximum $p$-spectral radius of a $k$-chromatic $3$-uniform hypergraph of order $n$ for $p \geq 1$?
• RQ4 Is the complete $k$-chromatic $3$-graph $Q_k^3(n)$ the unique maximizer of $\lambda^{(p)}(G)$ among $k$-chromatic $3$-graphs for $p \geq 1$?
• RQ5 Can the extremal bounds for $\lambda^{(p)}(G)$ be tightened and generalized beyond the known inequalities for $p=1$ and $p=2$?

Key findings

• For every $p > 1$, the $p$-spectral radius of any $k$-partite $r$-graph $G$ of order $n$ satisfies $\lambda^{(p)}(G) < \lambda^{(p)}(T_k^r(n))$, with equality if and only if $G = T_k^r(n)$, where $T_k^r(n)$ is the complete $k$-partite $r$-graph with parts of size $\lfloor n/k\rfloor$ or $\lceil n/k\rceil$.
• For every $p \geq 1$, the $p$-spectral radius of any $k$-chromatic $3$-graph $G$ of order $n$ satisfies $\lambda^{(p)}(G) < \lambda^{(p)}(Q_k^3(n))$, with equality if and only if $G = Q_k^3(n)$, the complete $k$-chromatic $3$-graph with balanced parts.
• The $p$-spectral radius $\lambda^{(p)}(G)$ is maximized uniquely by the complete $k$-partite hypergraph $T_k^r(n)$ among all $k$-partite $r$-graphs, confirming a conjecture for $r=3$ and extending it to general $r \geq 2$.
• The bound $\lambda^{(p)}(G) \leq 3! \left(\binom{n}{3} - k\binom{n/k}{3}\right) n^{-3/p}$ holds for $k$-chromatic $3$-graphs, with equality if and only if $G = Q_k^3(n)$ and $k \mid n$.
• The extremal $p$-spectral radius for $k$-chromatic $3$-graphs is strictly less than that of $Q_k^3(n)$ unless $G$ is isomorphic to $Q_k^3(n)$, generalizing a result of Mubayi and Talbot from $p=1$ to all $p \geq 1$.
• The results are tight and generalize classical extremal results for $p=1$ (Lagrangian) and $p=2$ (spectral radius), showing that the complete $k$-partite and $k$-chromatic hypergraphs are the unique maximizers across all $p \geq 1$.