Maximizing the spectral radius of generalized cactus graphs

For an integer $k\geq 0$, a connected graph $G$ is called a $k$-cactus graph if each edge $e \in E(G)$ 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.