Maximizing the spectral radius of generalized cactus graphs

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. Lov\'asz and Pelik\'an (1973), Borovi\'canin and Petrovi\'c (2006) 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.