  Synchronization and resonance on networks are some of the most remarkable
collective dynamical phenomena. The network topology, or the nature and
distribution of the connections within an ensemble of coupled oscillators,
plays a crucial role in shaping the local and global evolution of the two
phenomena. This article further explores this relationship within a compact
mathematical framework and provides new contributions on certain pivotal
issues, including a closed bound for the average synchronization time in
arbitrary topologies; new evidences of the effect of the coupling strength on
this time; exact closed expressions for the resonance frequencies in terms of
the eigenvalues of the Laplacian matrix; a measure of the effectiveness of an
\textit{influencer node}'s impact on the network; and, finally, a discussion on
the existence of a resonant synchronized state. Some properties of the solution
of the linear swing equation are also discussed within the same setting.
Numerical experiments conducted on two distinct real networks - a social
network and a power grid - illustrate the significance of these results and
shed light on intriguing aspects of how these processes can be interpreted
within networks of this kind.
