  We propose a graph-based approach to 5d superconformal field theories (SCFTs)
based on their realization as M-theory compactifications on singular elliptic
Calabi--Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d
$\mathcal{N}=(1,0)$ SCFTs by circle compactification and mass deformations. We
derive a description of these theories in terms of graphs, so-called Combined
Fiber Diagrams, which encode salient features of the partially resolved
Calabi--Yau geometry, and provides a combinatorial way of characterizing all 5d
SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly
capture strongly coupled data of the 5d SCFTs, such as the superconformal
flavor symmetry, BPS states, and mass deformations. The capabilities of this
approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The
full potential, however, becomes apparent when applied to theories with higher
rank. Starting with the higher rank conformal matter theories in 6d, we are led
to the discovery of previously unknown flavor symmetry enhancements and new 5d
SCFTs.
