On the spectral radius of bipartite graphs with given order and 4-independence number

College of Mathematics and System Science, Xinjiang University, Urumqi 830046, P.R. China

^* Corresponding author: ldxjedu@163.com

Received: 26 October 2024
Accepted: 3 June 2025

Abstract

For a positive integer k, let P_k be a path of order k. For a graph G, a subset S ⊆ V (G) is called a k-independent set if the induced subgraph G[S] does not contain P_k as its subgraph. The k-independence number of G, denoted by α_k(G), is the maximum cardinality of a k-independent set in G. Let B_n,αk be the set of bipartite graphs of order n with k-independence number α_k. What are the corresponding extremal graphs in B_n,αk with the maximum spectral radius? Lou and Guo [Discrete Math. 345 (2022) 112778] solved the problem for k = 2, and then Huang et al. [Discrete Appl. Math. 342 (2024) 368–380] provided the answer for k = 3. We are interested in the analogous problem for k = 4. In the paper, we determine the unique graph attaining the maximum spectral radius among all bipartite graphs in B_n,α4.