The first few unicyclic and bicyclic hypergraphs with larger spectral radii

TL;DR

This paper identifies the unicyclic and bicyclic hypergraphs with the largest spectral radii among all simple connected hypergraphs of these types, using edge operations and eigenvalue formulas.

Contribution

It determines the top hypergraphs with maximum spectral radii within unicyclic and bicyclic classes, introducing new methods for spectral analysis.

Findings

Identified the top five unicyclic hypergraphs with largest spectral radii.

Identified the top three bicyclic hypergraphs with largest spectral radii.

Developed new techniques using edge operations and eigenvalue formulas.

Abstract

A connected $k$-uniform hypergraph with $n$ vertices and $m$ edges is called $r$-cyclic if $n=m(k-1)-r+1$. For $r=1$ or $2$, the hypergraph is simply called unicyclic or bicyclic. In this paper we investigate hypergraphs that attain larger spectral radii among all simple connected $k$-uniform unicyclic and bicyclic hypergraphs. Specifically, by using some edge operations, the formula on power hypergraph eigenvalues, the weighted incidence matrix and a result on linear unicyclic hypergraphs, we determined the first five hypergraphs with larger spectral radius among all unicyclic hypergraphs and the first three over all bicyclic hypergraphs.