The effect on the spectral radius of r-graphs by grafting or contracting edges

TL;DR

This paper studies how grafting or contracting edges affects the spectral radius of r-uniform hypergraphs and provides an ordering of hypergraphs with minimal spectral radius for large sizes.

Contribution

It introduces a detailed analysis of spectral radius changes due to edge modifications in r-graphs and ranks hypergraphs with small spectral radius for large n.

Findings

Grafting or contracting edges significantly impacts the spectral radius.

Established an ordering of r-graphs with minimal spectral radius for n ≥ 20.

Provided theoretical bounds and comparisons for spectral radii in r-graphs.

Abstract

Let $\mathcal{H}^{(r)}_n$ be the set of all connected $r$-graphs with given size $n$. In this paper, we investigate the effect on the spectral radius of $r$-uniform hypergraphs by grafting or contracting an edge and then give the ordering of the $r$-graphs with small spectral radius over $\mathcal{H}^{(r)}_n$, when $n\geq 20$.