  The causal representation of multi-loop scattering amplitudes, obtained from
the application of the loop-tree duality formalism, comprehensively elucidates,
at integrand level, the behaviour of only physical singularities. This
representation is found to manifest compact expressions for multi-loop
topologies that have the same number of \textit{vertices}. Interestingly,
integrands considered in former studies, with up-to six vertices and $L$
internal lines, display the same structure of up-to four-loop ones. The former
is an insight that there should be a correspondence between vertices and the
collection of internal lines, \textit{edges}, that characterise a multi-loop
topology. By virtue of this relation, in this paper, we embrace an approach to
properly classify multi-loop topologies according to vertices and edges.
Differently from former studies, we consider the most general topologies, by
connecting vertices and edges in all possible ways. Likewise, we provide a
procedure to generate causal representation of multi-loop topologies by
considering the structure of causal propagators. Explicit causal
representations of loop topologies with up-to nine vertices are provided.
