Generalizations of some Nordhaus–Gaddum-type results on spectral radius

Let $G$ be a simple graph and $\lambda \left(G\right)$ the spectral radius of $G$. For $k\ge 2$, a $k$-edge decomposition $\left(H_1,\dots ,H_k\right)$ is $k$ spanning subgraphs such that their edge sets form a $k$-partition of the edge set of $G$. In this paper, we obtain some sharp lower and upper bounds for $\lambda \left(H_1\right)+\cdots +\lambda \left(H_k\right)$ in terms of the clique number of $H_i$ and the size of $G$, and discuss what $k$-edge decomposition $\left(H_1,\dots ,H_k\right)$ can maximize $\lambda \left(H_1\right)+\cdots +\lambda \left(H_k\right)$ when $G$ is a complete graph. These generalize some Nordhaus–Gaddum-type results on spectral radius for $k=2$, due to Nosal, Hong and Shu, and Nikiforov.