Fractional matching number and spectral radius of nonnegative matrices of graphs. Issue 19 (16th December 2022)

• Record Type: Journal Article
• Authors: Liu, Ruifang; Lai, Hong-Jian; Guo, Litao; Xue, Jie
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 19(2022)
• Page Start: 4133
• Page End: 4145
• Publication Date: 2022-12-16
• Subjects: Nonnegative matrix -- spectral radius -- fractional matching -- fractional perfect matching -- quotient matrix; 05C50; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1865252
• Languages: English
• ISSNs: 0308-1087

Abstract

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.