Fractional matching number and spectral radius of nonnegative matrices of graphs

ABSTRACT

A fractional matching of a graph G is a function f:E(G) → [0, 1] such that for any v ∈ V(G), $\sum _{e\in E_G\left(v\right)}f\left(e\right)\le 1$ where E_G(v) = {e ∈ E(G): e is incident with v in G}. The fractional matching number of G is ${\mu }_f\left(G\right)=max\left\{\sum _{e\in E\left(G\right)}f\right(e):f$ is a fractional matching of G}. For any real numbers a ≥ 0 and k ∈ (0, n), it is observed that if n = |V(G)| and $\delta \left(G\right)>\frac{n-k}{2}$, then ${\mu }_f\left(G\right)>\frac{n-k}{2}$. We determine a function φ(a, n, δ, k) and show that for a connected graph G with n = |V(G)|, $\delta \left(G\right)\le \frac{n-k}{2}$, spectral radius λ₁(G) and complement $\overline{G}$, each of the following holds.

1. If λ₁ (aD(G) + A(G)) < φ(a, n, δ, k), then ${\mu }_f\left(G\right)>\frac{n-k}{2}.$

2. If ${\lambda }_1\left(aD\right(\overline{G})+A(\overline{G}\left)\right)<(a+1)(\delta +k-1),$ then ${\mu }_f\left(G\right)>\frac{n-k}{2}.$

As applications, we prove a relationship between μ_f(G) and λ₁(aD(G) + A(G)) for a graph G. Furthermore, sufficient spectral conditions for a graph to have a fractional perfect matching are also obtained.

Keywords:

• Nonnegative matrix
• spectral radius
• fractional matching
• fractional perfect matching
• quotient matrix

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. The research of Ruifang Liu is supported by NSFC (No. 11971445) and NSF of Henan Province (No. 202300410377). The research of Jie Xue is supported by NSFC (No. 12001498) and China Postdoctoral Science Foundation (No. 2020M682325).

Disclosure statement

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

Additional information

Funding

The research of Ruifang Liu is supported by NSFC [grant number 11971445] and NSF of Henan Province [grant number 202300410377], and The research of Jie Xue is supported by NSFC [grant number 12001498] and China Postdoctoral Science Foundation [grant number 2020M682325].