Pressure type metrics on spaces of metric graphs

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil–Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the “classic” Weil–Petersson metric in Teichmüller theory. This paper is motivated by Pollicott and Sharp’s work (Pollicott and Sharp in Geom Dedic 172(1):229–244, 2014). Moreover, we fix some errors in Pollicott and Sharp (2014).