Pith. sign in

REVIEW 5 cited by

What is actually a metric graph?

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 1912.07549 v2 pith:EPD2WSWG submitted 2019-12-16 math.CO math.FAmath.GN

Delio Mugnolo This is my paper · ORCID
classification math.COmath.FAmath.GN
keywords graphmetrichereabstractactuallyalreadyassociatingcasually
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
read the original abstract

Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an abstract, self-contained definition of metric graph. Being mostly topological, it doesn't require any knowledge from graph theory and already determines uniquely several concepts that are commonly and unnecessarily \textit{defined} in the literature. Nevertheless, many ideas mentioned here are folklore in the quantum graph community: we discuss them for later reference.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. General diffusions on metric graphs as limits of time-space Markov Chains

    math.PR 2025-07 conditional novelty 7.0 of 10

    A new space-time Markov chain approximation for general diffusions on metric graphs is shown to converge in p-Wasserstein distance at explicit rates governed by a thinness quantifier of the subdivision.

  2. On the heat content of compact quantum graphs

    math.SP 2025-02 conditional novelty 7.0 of 10

    For compact metric graphs with Dirichlet conditions, the heat content is shown to equal the volume minus a boundary term plus a weighted sum over Dirichlet-to-Dirichlet paths, for all positive times.

  3. The role of expanders in the spectral geometry of metric graphs

    math.SP 2026-07 accept novelty 6.0 of 10

    Unilateral metric graphs built from Ramanujan expanders disprove geometric upper bounds on the spectral gap and show the Pólya–Szegő inequality is asymptotically sharp.

  4. On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs

    math.SP 2025-09 conditional novelty 5.0 of 10

    On tree quantum graphs, the n-th Laplacian eigenfunction has at most n-1 zeros, and under genericity assumptions the spectral minimal partition energy equals the (n+1)-th Neumann eigenvalue.

  5. Combinatorial properties of continuous graphs: A survey of challenges, solutions and open problems

    math.CO 2025-01 conditional novelty 4.0 of 10

    A survey of complexity results and open problems for independent set, vertex cover, coloring, and treewidth on continuous graphs, with a new upper bound for the coloring number of complete continuous graphs.

Pith tools