  In arXiv:1906.11820 and arXiv:1907.05404 we proposed an approach based on
graphs to characterize 5d superconformal field theories (SCFTs), which arise as
compactifications of 6d $\mathcal{N}= (1,0)$ SCFTs. The graphs, so-called
combined fiber diagrams (CFDs), are derived using the realization of 5d SCFTs
via M-theory on a non-compact Calabi--Yau threefold with a canonical
singularity. In this paper we complement this geometric approach by connecting
the CFD of an SCFT to its weakly coupled gauge theory or quiver descriptions
and demonstrate that the CFD as recovered from the gauge theory approach is
consistent with that as determined by geometry. To each quiver description we
also associate a graph, and the embedding of this graph into the CFD that is
associated to an SCFT provides a systematic way to enumerate all possible
consistent weakly coupled gauge theory descriptions of this SCFT. Furthermore,
different embeddings of gauge theory graphs into a fixed CFD can give rise to
new UV-dualities for which we provide evidence through an analysis of the
prepotential, and which, for some examples, we substantiate by constructing the
M-theory geometry in which the dual quiver descriptions are manifest.
