  We investigate the existence of ground states at prescribed mass on general
metric graphs with half-lines for focusing doubly nonlinear Schr\"odinger
equations involving both a standard power nonlinearity and delta nonlinearities
located at the vertices. The problem is proved to be sensitive both to the
topology and to the metric of the graph and to exhibit a phenomenology richer
than in the case of the sole standard nonlinearity considered in [Adami et al.
'15, Adami et al. '16]. On the one hand, we provide a complete topological
characterization of the problem, identifying various topological features
responsible for existence/non-existence of doubly nonlinear ground states in
specific mass regimes. On the other hand, we describe the role of the metric in
determining the exact interplay between these different topological properties.
