Fractional matching number and eigenvalues of a graph

ABSTRACT

A fractional matching of a graph G is a function f giving 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 of G, written ${\nu }^{\ast }\left(G\right)$, is the maximum of $\sum _{e\in E\left(G\right)}f\left(e\right)$ over all fractional matchings. In this paper, we study the connections between the fractional matching number and the Laplacian spectral radius of a graph. We also give some sufficient spectral conditions for the existence of a fractional perfect matching.

KEYWORDS:

• Eigenvalue
• Laplacian eigenvalue
• fractional matching
• fractional perfect matching

AMS CLASSIFICATION:

• 05C50

Acknowledgments

The authors would like to thank the anonymous referees very much for valuable suggestions and corrections which lead to a great improvement in the original paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by National Natural Science Foundation of China (No. 11471121) and Key Projects for Top-notch Talents of Anhui Province (No. JXBJZD2016082).