Maximizing the spectral radius of generalized cactus graphs

¹ School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian, P.R. China
² School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 October 2025
Accepted: 2 April 2026

Abstract

For an integer k ≥ 0, a connected graph G is called a k-cactus graph if each edge e ∈ E(G) is contained in at most k cycles of G. Inspired by the Brualdi-Solheid problem, in this paper, we address the problem of determining the maximum spectral radius of k-cactus graphs. Lovász and Pelikán [Period. Math. Hungar. 3 (1973) 175–182], Borovićanin and Petrović [Publ. Inst. Math. (Beograd) (N.S.) 79 (2006) 13–18] resolved the cases of k = 0 and k = 1, respectively. We solve this problem for the cases of k = 2 and k = 3 that is the graphs with the maximum spectral radius among all 2-cactus graphs and 3-cactus graphs are determined, respectively.