[Paper Review] Ordering uniform supertrees by their spectral radii

Xiying Yuan|arXiv (Cornell University)|Feb 17, 2015

Tensor decomposition and applications6 references4 citations

TL;DR

This paper orders $k$-uniform supertrees by their spectral radii using edge-moving operations and the weighted incidence matrix method, establishing the first four supertrees with the largest spectral radii among all $k$-uniform supertrees on $n$ vertices. It proves that the hyperstar $S_n^k$, the $k$-uniform hypertree $S^k(2,n'-4)$, the 3-vertex supertree $\mathcal{T}(1,1,m-3)$, and $S^k(3,n'-5)$ are the top four in spectral radius order, with strict inequalities for non-isomorphic structures.

ABSTRACT

A connected and acyclic hypergraph is called a supertree. In this paper we mainly focus on the spectral radii of uniform supertrees. Li, Shao and Qi determined the first two $k$-uniform supertrees with large spectral radii among all the $k$-uniform supertrees on $n$ vertices [H. Li, J. Shao, L. Qi, The extremal spectral radii of $k$-uniform supertrees, arXiv:1405.7257v1, May 2014]. By applying the operation of moving edges on hypergraphs and using the weighted incidence matrix method we extend the above order to the fourth $k$-uniform supertree.

Motivation & Objective

• To determine the ordering of $k$-uniform supertrees by their spectral radii, extending prior work that identified only the first two.
• To resolve the structure of the third and fourth largest spectral radii in $k$-uniform supertrees on $n$ vertices.
• To apply the weighted incidence matrix method and edge-moving operations to compare spectral radii across different supertree configurations.
• To establish strict inequalities in spectral radius for non-isomorphic supertrees, particularly distinguishing between hypertrees and general supertrees.

Proposed method

• Uses the adjacency tensor and spectral radius definition for $k$-uniform hypergraphs, where the spectral radius is the largest eigenvalue of the adjacency tensor.
• Applies the weighted incidence matrix method to analyze the spectral radius, defining a matrix $B$ with entries $B(v,e)$ for vertex-edge pairs.
• Employs edge-moving operations on hypergraphs to transform supertrees into configurations with larger spectral radii, using the monotonicity of the spectral radius under such operations.
• Uses the concept of $\alpha$-supernormality and proves strict inequalities via eigenvalue interlacing and tensor eigenequations.
• Leverages the principal eigenvector and weak irreducibility of the adjacency tensor to ensure positivity and uniqueness of the dominant eigenvector.
• Applies known results on power hypergraphs and hypertrees to characterize supertrees by their non-pendent vertex count and edge structure.

Experimental results

Research questions

• RQ1 What is the third largest spectral radius among all $k$-uniform supertrees on $n$ vertices, and which supertree achieves it?
• RQ2 What is the fourth largest spectral radius among $k$-uniform supertrees, and which structure realizes it?
• RQ3 How do edge-moving operations affect the spectral radius of $k$-uniform supertrees?
• RQ4 Can the weighted incidence matrix method be used to compare spectral radii of non-isomorphic $k$-uniform supertrees?
• RQ5 What is the relationship between the number of non-pendent vertices and the spectral radius in $k$-uniform supertrees?

Key findings

• The hyperstar $S_n^k$ has the largest spectral radius among all $k$-uniform supertrees on $n$ vertices.
• The $k$-uniform hypertree $S^k(2,n'-4)$ has the second-largest spectral radius, strictly greater than all other $k$-uniform supertrees except $S_n^k$.
• The supertree $\mathcal{T}(1,1,m-3)$, with three non-pendent vertices sharing one edge, has the third-largest spectral radius.
• The $k$-uniform hypertree $S^k(3,n'-5)$ has the fourth-largest spectral radius, strictly less than $\mathcal{T}(1,1,m-3)$ but greater than all other non-isomorphic configurations.
• For any $k$-uniform supertree $\mathcal{T}$ not isomorphic to $S_n^k$, $S^k(2,n'-4)$, $\mathcal{T}(1,1,m-3)$, or $S^k(3,n'-5)$, its spectral radius is strictly less than that of $\mathcal{T}(1,1,m-3)$.
• All spectral radius inequalities are strict and proven via the weighted incidence matrix method and $\alpha$-supernormality, ensuring no ties among non-isomorphic structures.