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In-phase synchronization is a stable state of identical Kuramoto oscillatorscoupled on a network with identical positive connections, regardless of networktopology. However, this fact does not mean that the networks always synchronizein-phase because other attractors besides the stable state may exist. Thecritical connectivity $\mu_{\mathrm{c}}$ is defined as the network connectivityabove which only the in-phase state is stable for all the networks. In otherwords, below $\mu_{\mathrm{c}}$, one can find at least one network which has astable state besides the in-phase sync. The best known evaluation of the valueso far is $0.6828\cdots\leq\mu_{\mathrm{c}}\leq0.75$. In this paper, focusingon the twisted states of the circulant networks, we provide a method tosystematically analyze the linear stability of all possible twisted states onall possible circulant networks. This method using integer programming enablesus to find the densest circulant network having a stable twisted state besidesthe in-phase sync, which breaks a record of the lower bound of the$\mu_{\mathrm{c}}$ from $0.6828\cdots$ to $0.6838\cdots$. We confirm thevalidity of the theory by numerical simulations of the networks not convergingto the in-phase state.

The lower bound of the network connectivity guaranteeing in-phase synchronization