ABSTRACT
A fractional matching of a graph G is a function f:E(G) → [0, 1] such that for any v ∈ V(G), where EG(v) = {e ∈ E(G): e is incident with v in G}. The fractional matching number of G is
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
, then
. We determine a function φ(a, n, δ, k) and show that for a connected graph G with n = |V(G)|,
, spectral radius λ1(G) and complement
, each of the following holds.
If λ1 (aD(G) + A(G)) < φ(a, n, δ, k), then
If
then
AMS Classification:
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).