[Paper Review] The extremal spectral radii of $k$-uniform supertrees

Honghai Li, Jia‐Yu Shao+1 more|arXiv (Cornell University)|May 28, 2014

Tensor decomposition and applications16 references17 citations

TL;DR

This paper investigates the extremal spectral radii of $k$-uniform supertrees using hypergraph perturbation techniques. It proves that the hyperstar $\mathcal{S}_{n,k}$ uniquely maximizes the adjacency, signless Laplacian, and incidence $Q$-spectral radii among all $k$-uniform supertrees on $n$ vertices, while the loose path $\mathcal{P}_{n,k}$ uniquely minimizes the spectral radius among $k$-th power hypertrees.

ABSTRACT

In this paper, we study some extremal problems of three kinds of spectral radii of $k$-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence $Q$-spectral radius). We call a connected and acyclic $k$-uniform hypergraph a supertree. We introduce the operation of "moving edges" for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar $\mathcal{S}_{n,k}$ attains uniquely the maximum spectral radius among all $k$-uniform supertrees on $n$ vertices. We also determine the unique $k$-uniform supertree on $n$ vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path $\mathcal{P}_{n,k}$ attains uniquely the minimum spectral radius among all $k$-th power hypertrees of $n$ vertices. Some bounds on the incidence $Q$-spectral radius are given. The relation between the incidence $Q$-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.

Motivation & Objective

• To determine the $k$-uniform supertrees that extremize three types of spectral radii: adjacency, signless Laplacian, and incidence $Q$-spectral radius.
• To establish extremal structures—hyperstar and loose path—among $k$-uniform supertrees and $k$-th power hypertrees.
• To develop and apply perturbation operations—edge-releasing and total grafting—to analyze spectral radius changes under structural modifications.
• To derive bounds on the incidence $Q$-spectral radius and clarify its relationship with the spectral radius of the matrix product $RR^T$.

Proposed method

• Introduced the 'moving edges' operation as a general framework, with two special cases: edge-releasing and total grafting.
• Proved that edge-releasing and inverse total grafting strictly increase all three spectral radii, enabling extremal structure identification.
• Used tensor eigenvalue theory, including adjacency, signless Laplacian, and incidence $Q$-tensors, to analyze spectral radii.
• Applied Jensen’s inequality and H"older-type inequalities to derive bounds on the incidence $Q$-spectral radius.
• Established a spectral radius relationship: $\rho(RR^T) < \rho(\mathcal{Q}^*) < k^{k-2}\rho(RR^T)$, linking matrix and tensor spectral radii.
• Leveraged nonnegative irreducibility and Perron-Frobenius theory for nonnegative tensors to analyze eigenvectors and equality conditions.

Experimental results

Research questions

• RQ1 Which $k$-uniform supertree maximizes the adjacency, signless Laplacian, and incidence $Q$-spectral radii?
• RQ2 Which $k$-uniform supertree has the second-largest spectral radius among all $k$-uniform supertrees for these three spectral radii?
• RQ3 Which $k$-th power hypertree minimizes the spectral radius among all such hypertrees for these three spectral radii?
• RQ4 How do the spectral radii of the incidence $Q$-tensor relate to those of the matrix product $RR^T$?
• RQ5 What are tight upper and lower bounds for the incidence $Q$-spectral radius in $k$-uniform hypergraphs?

Key findings

• The hyperstar $\mathcal{S}_{n,k}$ uniquely attains the maximum spectral radius among all $k$-uniform supertrees on $n$ vertices for all three spectral radii.
• The loose path $\mathcal{P}_{n,k}$ uniquely attains the minimum spectral radius among all $k$-th power hypertrees on $n$ vertices for all three spectral radii.
• For the incidence $Q$-spectral radius, the bound $\rho(\mathcal{Q}^*) \leq k^{k-1}\Delta$ holds, with equality if and only if the hypergraph is $\Delta$-regular.
• The spectral radius of the incidence $Q$-tensor satisfies $\rho(RR^T) < \rho(\mathcal{Q}^*) < k^{k-2}\rho(RR^T)$, providing a tight comparison with the matrix product spectral radius.
• The edge-releasing operation and inverse total grafting operation strictly increase all three spectral radii, enabling extremal structure characterization.
• The hyperstar is the unique maximizer and the loose path the unique minimizer across all three spectral radii, demonstrating structural universality in extremal hypergraph spectral theory.