The minimum spectral radius of the r-uniform supertree having two vertices of maximum degree

Abstract

A supertree is a connected and acyclic hypergraph. Let ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$ be the set of the r-uniform supertrees with m edges and exactly two vertices of maximum degree ▵, where $m\ge 5$ and $3\le \mathrm{△}\le \left[\right(m+1\left)/2\right]$. We investigate the supertree with the minimum spectral radius among ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$. Several useful grafting operations are introduced for comparing the spectral radii of supertrees. The supertree with the minimum spectral radius among ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$ is characterized. In addition, a new simple method is demonstrated to obtain the supertree with the minimum spectral radius among the set of r-uniform supertrees with m edges.

Keywords:

• Hypergraph
• spectral radius
• supertree
• matching polynomial

2010 Mathematics Subject Classifications:

• 05C35
• 05C65

Acknowledgments

The work was supported by the National Natural Science Foundation of China under the grant numbers 11871040 and 11001166. The author thanks the anonymous reviewer for his/her helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China under the grant numbers 11871040 and 11001166.