On the distance spectral radius, fractional matching and factors of graphs with given minimum degree

¹ College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P.R. China
² School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, P.R. China

^* Corresponding author: xiyanxwg@163.com

Received: 20 January 2025
Accepted: 6 July 2025

Abstract

A fractional matching of G is a function f : E(G) → [0, 1] such that ∑e∈E_G(v_i) f(e) ≤ 1 for any v_i ∈ V (G), where E_G(v_i) = {e : e ∈ E(G) and e is incident with v_i}. Let α_f (G) denote the fractional matching number of G, which is defined as α_f (G) = max{∑e∈E(G) f(e) : f is a fractional matching of G}. Let {G₁, G₂, G₃, . . .} be a set of graphs, a {G₁, G₂, G₃, . . .}-factor of a graph G is a spanning subgraph of G such that each component of which is isomorphic to one of {G₁, G₂, G₃, . . .}. In this paper, we first establish a sharp upper bound for the distance spectral radius to guarantee that α_f (G) > n−k/2 in a graph G of order n with given minimum degree, where 0 < k < n is an integer. Then we give a sharp upper bound on the distance spectral radius of a graph G with given minimum degree δ to ensure that G has a {K₂, {C_k}}-factor, where 3 ≤ k < +∞ is an integer. Moreover, we obtain a sharp upper bound on the distance spectral radius for the existence of a {K_1,1, K_1,2, . . . , K_1,k}-factor with 2 ≤ k < +∞ in a graph G with given minimum degree.