Generalizations of some Nordhaus–Gaddum‐type results on spectral radius.

Let G $G$ be a simple graph and λ(G) $\lambda (G)$ the spectral radius of G $G$. For k≥2 $k\ge 2$, a k $k$‐edge decomposition (H1,...,Hk) $({H}_{1},{\rm{\ldots }},{H}_{k})$ is k $k$ spanning subgraphs such that their edge sets form a k $k$‐partition of the edge set of G $G$. In this paper, we obtain some sharp lower and upper bounds for λ(H1)+⋯+λ(Hk) $\lambda ({H}_{1})+\,\cdots \,+\lambda ({H}_{k})$ in terms of the clique number of Hi ${H}_{i}$ and the size of G $G$, and discuss what k $k$‐edge decomposition (H1,...,Hk) $({H}_{1},{\rm{\ldots }},{H}_{k})$ can maximize λ(H1)+⋯+λ(Hk) $\lambda ({H}_{1})+\cdots \,+\lambda ({H}_{k})$ when G $G$ is a complete graph. These generalize some Nordhaus–Gaddum‐type results on spectral radius for k=2 $k=2$, due to Nosal, Hong and Shu, and Nikiforov. [ABSTRACT FROM AUTHOR]