Fractional matching number and spectral radius of nonnegative matrices of graphs. Issue 19 (16th December 2022) - Liu, Ruifang; Lai, Hong-Jian; Guo, Litao; Xue, Jie
ABSTRACT: A fractional matching of a graph G is a function f : E ( G ) → [0, 1] such that for any v  ∈  V ( G ), ∑ e ∈ E G ( v ) f ( e ) ≤ 1 where E G ( v ) = { e  ∈  E ( G ): e is incident with v in G }. The fractional matching number of G is μ f ( G ) = max { ∑ e ∈ E ( G ) f ( 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 δ ( G ) > n − k 2, then μ f ( G ) > n − k 2 . We determine a function φ( a, n, δ, k ) and show that for a connected graph G with n  = | V ( G )|, δ ( G ) ≤ n − k 2, spectral radius λ 1 ( G ) and complement G ¯, each of the following holds. If λ 1 ( aD ( G ) +  A ( G )) < φ( a, n, δ, k ), then μ f ( G ) > n − k 2 . If λ 1 ( a D ( G ¯ ) + A ( G ¯ ) ) < ( a + 1 ) ( δ + k − 1 ), then μ f ( G ) > n − k 2 . As applications, we prove a relationship between μ f ( G ) and λ 1 ( aD ( G ) +  A ( G )) for a graph G . Furthermore, sufficient spectral conditions for a graph to have a fractional perfect matching are also obtained.