The matching polynomials and spectral radii of uniform supertrees

TL;DR

This paper investigates the spectral radii of uniform supertrees by analyzing their matching polynomials, extending graph grafting results, and identifying extremal spectral radii for various supertree configurations.

Contribution

It extends grafting operation techniques from graphs to supertrees and determines extremal spectral radii for uniform supertrees based on size and diameter.

Findings

Identifies the top  largest spectral radii for supertrees with given size and diameter.

Determines the two smallest spectral radii among supertrees of a given size.

Extends graph grafting methods to hypergraph structures.

Abstract

We study matching polynomials of uniform hypergraph and spectral radii of uniform supertrees. By comparing the matching polynomials of supertrees, we extend Li and Feng's results on grafting operations on graphs to supertrees. Using the methods of grafting operations on supertrees and comparing matching polynomials of supertrees, we determine the first $\lfloor\frac{d}{2}\rfloor+1$ largest spectral radii of $r$-uniform supertrees with size $m$ and diameter $d$. In addition, the first two smallest spectral radii of supertrees with size $m$ are determined.