Fractional matching, factors and spectral radius in graphs involving minimum degree

A fractional matching of a graph G is a function f : E(G) -> [0, 1] such that for any v is an element of V (G), sigma e is an element of EG(v) f (e) <= 1, where EG(v) = {e is an element of E(G) : e is incident with v in G}. The frac-tional matching number of G is mu f (G) = max{sigma e is an element of E(G) f (e) : f is a fractional matching of G}. Let k is an element of (0, n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee mu f(G) > n-k 2 in a graph with minimum degree delta, which implies the result on the fractional perfect matching due to Fan et al. (2022) [6]. For a set {A, B, C, ... } of graphs, an {A, B, C, ... }-factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of {A, B, C, ...}. We present a tight sufficient condition in terms of the spectral radius for the existence of a {K2, {Ck}}-factor in a graph with minimum degree delta, where k > 3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a {K1,1, K1,2, . . . , K1,k}-factor with k > 2 in a graph with minimum degree delta, which generalizes the result of Miao et al. (2023) [10].(c) 2023 Elsevier Inc. All rights reserved.