This paper studies real-time gross settlement (RTGS), which is a trend in interbank settlement systems. We provide a theoretical analysis of settlement efficiencyby focusing on the importance of the interconnected features of underlying payment networks. Specifically, we develop a graph-theoretic model to investigate a general class of payment networks, and examine the network factors that contribute to the lower bound and upper bound of the required settlement fund. The lower bound and upper bound are considered through a relevant minimization/maximization problem formulated on a weighted multi-arrow directed graph. We succeed in characterizing the lower bound and upper bound with two kinds of network factors: domain factor and synchronization factor. For given obligations, each domain refers to how far one unit of a settlement fund can circulate to settle the obligations, and the synchronization factor refers to how those domains are mutually consistent in the dynamics of the settlement. The main contribution of this study is to reveal the qualitative aspect of the synchronization factor with regard to two original concepts (arrow-twisted and vertex-twisted) that are formally defined on a directed graph. We further examine the quantitative aspect of the synchronization factor for realworld payment networks, through examining networks with clustered structures and small-world structures. We show that increase of obligations has non-linear effect on the required settlement fund through the synchronization factor.