Definition of graph-dependent measures for arbitrary graph topologies

Develop a formal definition of the graph-dependent measure associated with arbitrary graphs, not only series-parallel graphs, in a way that extends the series-parallel construction and accommodates the absence of a simple recursive decomposition.

Background

The paper defines graph-dependent probability measures recursively for series-parallel graphs. The recursive construction uses series and parallel decompositions and is connected to free additive and multiplicative convolutions; for such graphs, the measure describes the properly rescaled limiting eigenvalue distribution associated with a random tensor network.

The paper notes that the same recursive relation formally remains relevant for general graphs, but arbitrary graphs need not decompose into trivial graphs through series and parallel operations. Consequently, the existing construction does not provide a formal measure for arbitrary graph topologies, and the authors state this as a significant open question requiring new techniques or frameworks.

References

A significant open question is how to formally define the graph-dependent measures for arbitrary graphs.

— Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks  (2506.23894 - Hu et al., 30 Jun 2025) in Section 4, subsection “Relation to random tensor networks”

However, this viewpoint does not immediately extend to more complicated symbols or to operators involving multiple creation and annihilation terms, where the underlying graph structure is no longer composed of simple linear clusters and the combinatorics become substantially more intricate. We leave this problem for future investigation.

— Limiting eigenvalue distribution and entropy of multi-Toeplitz matrices  (2608.17859 - Jury et al., 18 Aug 2026) in Section 5, “Random tridiagonal matrices”