On the irregularity of uniform hypergraphs

TL;DR

This paper investigates the irregularity of r-uniform hypergraphs by analyzing spectral radius differences, degree deviations, and related measures, extending known results from graphs to hypergraphs.

Contribution

It introduces new bounds and relationships for hypergraph irregularity measures, extending spectral and degree-based results from graphs to hypergraphs.

Findings

Bounds on spectral radius deviations in hypergraphs

Relationships between degree irregularity measures

Extension of graph irregularity results to hypergraphs

Abstract

Let $H$ be an $r$-uniform hypergraph on $n$ vertices and $m$ edges, and let $d_i$ be the degree of $i\in V(H)$. Denote by $\varepsilon(H)$ the difference of the spectral radius of $H$ and the average degree of $H$. Also, denote \[ s(H)=\sum_{i\in V(H)}\left|d_i-\frac{rm}{n}\right|,~ v(H)=\frac{1}{n}\sum_{i\in V(H)}d_i^{\frac{r}{r-1}}-\left(\frac{rm}{n}\right)^{\frac{r}{r-1}}. \] In this paper, we investigate the irregularity of $r$-uniform hypergraph $H$ with respect to $\varepsilon(H)$, $s(H)$ and $v(H)$, which extend relevant results to uniform hypergraphs.