Distance spectral radius and fractional matching in t-connected graphs

Abstract

A fractional matching of a graph G is a function f assigning each edge a number in $[0,1]$ so that $\sum _{e\in \mathrm{\Gamma }\left(v\right)}f\left(e\right)\le 1$ for each $v\in V\left(G\right)$, where $\mathrm{\Gamma }\left(v\right)$ is the set of edges incident to v. The fractional matching number is the maximum of $\sum _{e\in \mathrm{\Gamma }\left(v\right)}f\left(e\right)$ over all fractional matchings. Motivated by progress in the study of relations between eigenvalues and matchings of graphs, in this paper, we characterize graphs with the minimum distance spectral radius among all t-connected graphs with n vertices and fractional matching number at most $\frac{n-k}{2}$ for $1\le k\le n-2$. Our characterization generalizes a result of Li, Miao, and Zhang [On the size, spectral radius, distance spectral radius and fractional matchings in graphs. Bull Aust Math Soc. 2023;187–199.], giving a distance spectral condition for the existence of a fractional perfect matching in a connected graph.

Keywords:

• Distance spectral radius
• fractional matching
• connectivity

AMS classification(2020):

• 05C50
• 05C35

Acknowledgments

The authors would like to show great gratitude to the editor and anonymous reviewers for their helpful suggestions and comments, which led to an improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).