2020/02/02 by Ruifang Liu, Hong‐Jian Lai, Liu, Ruifang +5
Mathematics · Computer Science · #Graph theory and applications #Matrix Theory and Algorithms #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.2002.00370
A fractional matching of a graph G is a function f:E(G) → [0,1] such that for any v∈ V(G), ∑e∈ EG(v)f(e)≤ 1 where EG(v) = \e ∈ E(G): e is incident with v in G\. The fractional matching number of G is μf(G) = max\∑e∈ E(G) f(e): f is 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 μf(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 λ1(G) and complement G, each of the following holds. (i) If λ1(aD(G)+A(G))(n-k)/(2). (ii) If λ1(aD(G)+A(G))(n-k)/(2). As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving Q-index and Aα-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.