A distance spectral radius for star-factors with large components in graphs

¹ School of Public Management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212100, P.R. China
² School of Economics and Management, Wenzhou University of Technology, Wenzhou, Zhejiang 325000, P.R. China

^* Corresponding author: zw_wzu@163.com

Received: 13 October 2024
Accepted: 20 June 2025

Abstract

Let G be a graph and let m be an integer with m ≥ 2. A {K_1,j : m ≤ j ≤ 2m}-factor of a graph G is a spanning subgraph of G, in which every component is isomorphic to an element in {K_1,j : m ≤ j ≤ 2m}. For v_i, v_j ∈ V (G), the distance between v_i and v_j, denoted by d_G(v_i, v_j) (or d_ij), is the length of a shortest path from v_i to v_j. Particularly, d_G(v_i, v_i) = 0 (or d_ii = 0). The distance matrix of G, denoted by D(G), is the n × n real symmetric matrix in which (i, j)-entry is d_G(v_i, v_j) (or d_ij). The largest eigenvalue of D(G), denoted by μ(G), is called the distance spectral radius of G. In this paper, it is proved that a connected graph G of order n ≥ 3m² + 6m + 2 with μ(G) ≤ μ(K_m−1 ∨ (K_n−m ∪ K₁)) contains a {K_1,j : m ≤ j ≤ 2m}-factor unless G = K_m−1 ∨ (K_n−m ∪ K₁).