Lyapunov equations that depend on communication graph topologies provide building blocks of Lyapunov functions, which play an important role in controller design and stability analysis of multi-agent systems. However, construction of such a Lyapunov equation in some published works has a flaw or requires unnecessarily strong graph conditions. This paper presents several choices of Lyapunov equations over various graph topologies. (C) 2015 Elsevier Ltd. All rights reserved.