  In-phase synchronization is a stable state of identical Kuramoto oscillators
coupled on a network with identical positive connections, regardless of network
topology. However, this fact does not mean that the networks always synchronize
in-phase because other attractors besides the stable state may exist. The
critical connectivity $\mu_{\mathrm{c}}$ is defined as the network connectivity
above which only the in-phase state is stable for all the networks. In other
words, below $\mu_{\mathrm{c}}$, one can find at least one network which has a
stable state besides the in-phase sync. The best known evaluation of the value
so far is $0.6828\cdots\leq\mu_{\mathrm{c}}\leq0.75$. In this paper, focusing
on the twisted states of the circulant networks, we provide a method to
systematically analyze the linear stability of all possible twisted states on
all possible circulant networks. This method using integer programming enables
us to find the densest circulant network having a stable twisted state besides
the 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 the
validity of the theory by numerical simulations of the networks not converging
to the in-phase state.
