{
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  "title": "Aletheia Technologies research publications",
  "home_page_url": "https://aletheia-technologies.it/en/research/",
  "feed_url": "https://aletheia-technologies.it/research/feed.json",
  "language": "en",
  "items": [
    {
      "id": "https://doi.org/10.5281/zenodo.21744051",
      "url": "https://aletheia-technologies.it/en/research/line-graph-inertia-roses-generalized-theta/",
      "external_url": "https://zenodo.org/records/21744051",
      "title": "Line-graph inertia of roses and generalized theta graphs",
      "content_text": "For a graph G, the adjacency inertia of the line graph L(G) is determined by the number of eigenvalues of the signless Laplacian Q(G) above, equal to, and below 2. The manuscript computes the inertia of Q(G) − 2I, and hence the inertia of the adjacency matrix of L(G), exactly, including every singular case, for rose graphs and generalized theta graphs. Both computations follow from a common reduction: deleting the common vertices leaves disjoint paths, and range-kernel elimination leaves their singular kernel directions and a residual scalar for a rose graph, or a 2 by 2 matrix for a generalized theta graph. The resulting formulas depend only on the path lengths modulo 4. Consequences include the bound m_Q(G,2) ≤ c(G) for generalized theta graphs with at least three paths and an exact comparison with the conjectured bound 2s(L(G)) ≤ c(G) + 1, whose slack grows linearly with the cyclomatic number on both classes, with equality only for cycles whose length is congruent to 1 modulo 4. The general conjecture is not proved. A partial extension to bridgeless cacti is also given, and exact finite computations check every formula branch.",
      "summary": "For two families of graphs, roses and generalized theta graphs, the work computes in closed form how many eigenvalues of the line graph are positive, zero and negative. The result depends only on the lengths of the paths, counted modulo four.",
      "date_published": "2026-08-01T00:00:00Z",
      "date_modified": "2026-08-01T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        },
        {
          "name": "Marco Paone",
          "url": "https://orcid.org/0009-0001-6792-879X"
        }
      ],
      "tags": [
        "rose graph",
        "generalized theta graph",
        "line graph",
        "adjacency inertia",
        "graph signature",
        "signless Laplacian",
        "generalized Schur complement",
        "spectral graph theory",
        "2020 MSC 05C50",
        "2020 MSC 15A18"
      ],
      "attachments": [
        {
          "url": "https://zenodo.org/records/21744051/files/Paone-Paone_Inertia-of-Line-Graphs-of-Rose-and-Generalized-Theta-Graphs_v1.0.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21744051/files/Paone-Paone_Inertia-of-Line-Graphs-of-Rose-and-Generalized-Theta-Graphs_source-and-reproducibility-v1.0.zip?download=1",
          "mime_type": "application/zip",
          "title": "Zenodo source and reproducibility package"
        }
      ]
    },
    {
      "id": "https://doi.org/10.5281/zenodo.21706797",
      "url": "https://aletheia-technologies.it/en/research/line-graph-signature-beyond-the-2-core/",
      "external_url": "https://zenodo.org/records/21706797",
      "title": "Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number",
      "content_text": "This preprint studies the signature of the adjacency matrix of line graphs using the shifted signless Laplacian Q(G) − 2I, rooted-response methods, and reductions to the 2-core. It proves an exact pendant-forest reduction, a parity property for rooted-tree responses, a singular attachment lemma in the graph-port setting, and a rank-one criterion determining when a pendant leaf increases the line-graph signature. It establishes the universal upper bound s(L(G)) ≤ c(G): together with the constructive lower bound this gives ⌊(c + 1) / 2⌋ ≤ f(c) ≤ c, so f(c) is finite and its maximum is attained for every cyclomatic number c. The sharper inequality 2s(L(G)) ≤ c(G) + 1 remains a conjecture. The load-bearing mathematical certificates and principal computational results were independently checked.",
      "summary": "An upper bound that holds for every graph, constructions that reach it, and counterexamples to a simplification that looked natural. Every case is checked with exact arithmetic, not approximation.",
      "date_published": "2026-07-30T00:00:00Z",
      "date_modified": "2026-07-30T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        },
        {
          "name": "Marco Paone",
          "url": "https://orcid.org/0009-0001-6792-879X"
        }
      ],
      "tags": [
        "line graph",
        "graph signature",
        "graph inertia",
        "spectral graph theory",
        "signless Laplacian",
        "cyclomatic number",
        "2-core",
        "rooted response",
        "Schur complement",
        "pendant forest",
        "extremal graph theory",
        "exact computation"
      ],
      "attachments": [
        {
          "url": "https://zenodo.org/records/21706797/files/line_graph_signature_beyond_2core_v1.3.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21706797/files/BEYOND_2CORE_PUBLIC_REPRODUCIBILITY_PACKAGE_v1.3.zip?download=1",
          "mime_type": "application/zip",
          "title": "Zenodo reproducibility package"
        }
      ]
    },
    {
      "id": "https://doi.org/10.5281/zenodo.21737348",
      "url": "https://aletheia-technologies.it/en/research/unbounded-signature-line-graphs/",
      "external_url": "https://zenodo.org/records/21737348",
      "title": "Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles",
      "content_text": "Akbari, Elphick, Kumar, Pragada and Tang conjectured that every connected graph satisfies a one-unit upper bound between the positive and negative adjacency inertia indices of its line graph. The manuscript presents a connected simple counterexample with line-graph inertia (9, 0, 7) and proves that the conjecture fails without bound even for connected simple planar subcubic cactus graphs. It introduces a rooted-module attachment lemma and a rooted C4-C5 signature amplifier, then develops an integral unimodular four-subdivision congruence preserving the determinant, adjacency cokernel, nonunit Smith factors, and nullity over every field. Two independent exact methods classify all 256 residue classes of three-cycle chains and agree row by row.",
      "summary": "Not one isolated counterexample but an infinite family, inside a class of very simple graphs, with the means to carry the result from one graph to another and exact certificates for every case.",
      "date_published": "2026-08-01T00:00:00Z",
      "date_modified": "2026-08-01T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        }
      ],
      "tags": [
        "line graph",
        "graph signature",
        "graph inertia",
        "inertia indices",
        "spectral graph theory",
        "cactus graph",
        "planar subcubic graph",
        "rooted module",
        "edge subdivision",
        "integral unimodular congruence",
        "Smith normal form",
        "adjacency cokernel",
        "exact residue classification"
      ],
      "attachments": [
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          "url": "https://zenodo.org/records/21737348/files/unbounded_signature_line_graphs_v2.0-rev2.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21737348/files/V2_PUBLIC_REPRODUCIBILITY_PACKAGE_v2.0-rev2.zip?download=1",
          "mime_type": "application/zip",
          "title": "Zenodo reproducibility package"
        }
      ]
    },
    {
      "id": "https://doi.org/10.5281/zenodo.21499790",
      "url": "https://aletheia-technologies.it/en/research/a-counterexample-line-graph-inertia-conjecture/",
      "external_url": "https://zenodo.org/records/21499790",
      "title": "A Counterexample to a Line-Graph Inertia Conjecture",
      "content_text": "The work presents a connected simple graph whose line graph has adjacency inertia (9, 0, 7), giving an exact counterexample to Conjecture 4.12. The public record includes an adjacency matrix, exact-arithmetic certificates, and independent scripts reproducing the inertia computation and incidence congruence.",
      "summary": "A finite connected simple counterexample to Conjecture 4.12 on line-graph inertia indices, accompanied by exact certificates.",
      "date_published": "2026-07-22T00:00:00Z",
      "date_modified": "2026-07-29T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        }
      ],
      "tags": [
        "line graph",
        "graph inertia",
        "inertia indices",
        "spectral graph theory",
        "counterexample",
        "Conjecture 4.12",
        "adjacency matrix",
        "characteristic polynomial",
        "exact arithmetic",
        "incidence congruence",
        "reproducibility certificate"
      ],
      "attachments": [
        {
          "url": "https://zenodo.org/records/21499790/files/Paone_2026_Line_Graph_Inertia_Counterexample_v1.0.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21499790/files/Paone_2026_Line_Graph_Inertia_Reproducibility_v1.0.zip?download=1",
          "mime_type": "application/zip",
          "title": "Zenodo reproducibility package"
        }
      ]
    },
    {
      "id": "https://doi.org/10.5281/zenodo.21810205",
      "url": "https://aletheia-technologies.it/en/research/from-rules-to-resilience/",
      "external_url": "https://zenodo.org/records/21810205",
      "title": "From Rules to Resilience: Assessing the EU's 2026 Action Plan on Cybersecurity and Artificial Intelligence",
      "content_text": "This Rapid Strategic Assessment asks whether the European Union can convert a regulator's legal power to obtain access to advanced AI models into a usable capability to evaluate them, and to defend critical infrastructure against AI-enabled cyber threats. The method separates what the Union inherited from what the 2026 Action Plan on Cybersecurity and Artificial Intelligence introduced, and assigns each proposition to an evidence stage: policy design, institutionally reported implementation, independently corroborated implementation, or observed performance. The central finding is a conversion problem. Union law had the access power and the published procedure; the reviewed public record had no completed evaluation, no allocated protected compute, no published methodology and no repeatedly usable service. The Plan's contribution is to sequence the missing functions around a baseline it inherited. The research cut-off is 5 August 2026.",
      "summary": "On 2 August 2026 the Commission gained the power to compel access to an advanced AI model for evaluation. This assessment asks what it can do with it, and finds the operational side of that power still unevidenced.",
      "date_published": "2026-08-05T00:00:00Z",
      "date_modified": "2026-08-05T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        }
      ],
      "tags": [
        "European Union",
        "artificial intelligence",
        "AI security",
        "cybersecurity",
        "critical infrastructure",
        "cyber policy",
        "ENISA",
        "NIS2",
        "Cyber Solidarity Act",
        "AI Act",
        "implementation readiness",
        "auditability"
      ],
      "attachments": [
        {
          "url": "https://zenodo.org/records/21810205/files/Paone_From-Rules-to-Resilience_v2.0.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21810205/files/From_Rules_to_Resilience_Supplementary_Evidence.zip",
          "mime_type": "application/zip",
          "title": "Supplementary evidence package"
        }
      ]
    },
    {
      "id": "https://doi.org/10.5281/zenodo.21793638",
      "url": "https://aletheia-technologies.it/en/research/response-protection-line-graph-equality-families/",
      "external_url": "https://zenodo.org/records/21793638",
      "title": "Response Protection for Line-Graph Equality Families: Transfer under Edge Subdivision and Rooted Attachment",
      "content_text": "Let L(G) denote the line graph of a connected graph G, and let c(G) = |E(G)| − |V(G)| + 1 be its cyclomatic number. Motivated by the open bound 2 sig(L(G)) ≤ c(G) + 1, the authors study how the line-graph signature changes when a missing edge is added. Using the rank-one edge-response criterion for M(G) = Q(G) − 2I, they formulate a four-inequality condition that prevents the relevant quadratic response from crossing the threshold at which an edge addition can increase the signature. The rank-one criterion and this threshold have direct antecedents; the new question addressed here is whether a closed family of response bounds survives natural graph operations. They prove that the condition is preserved by arbitrary-edge four-subdivision and by attaching a rooted C4-C5 module at an arbitrary vertex. Starting from C5, these operations generate an infinite class of connected planar cactus graphs attaining 2 sig(L(G)) = c(G) + 1. Every one-edge extension satisfies the same bound, and a general rank-one step yields a two-edge corollary. The proofs combine Schur complements with complete finite exact local checks. The response condition is sufficient rather than known to be necessary, and the universal cyclomatic bound remains open.",
      "summary": "Adding an edge to a graph can push a quantity tied to its line graph past a threshold. The paper gives a condition that prevents that jump, and proves the condition survives two ways of growing the graph: splitting an edge into four, and attaching a module at any vertex. Starting from a pentagon, this builds an infinite family of graphs sitting exactly on the bound. The condition is enough, but it is not known to be necessary, and the general bound stays open.",
      "date_published": "2026-08-04T00:00:00Z",
      "date_modified": "2026-08-04T00:00:00Z",
      "authors": [
        {
          "name": "Andrea Paone",
          "url": "https://orcid.org/0009-0003-6194-948X"
        },
        {
          "name": "Marco Paone",
          "url": "https://orcid.org/0009-0001-6792-879X"
        }
      ],
      "tags": [
        "graph inertia",
        "line graph",
        "signless Laplacian",
        "response protection",
        "edge addition",
        "edge subdivision",
        "cactus graph",
        "Schur complement",
        "spectral graph theory",
        "2020 MSC 05C50",
        "2020 MSC 15A18"
      ],
      "attachments": [
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          "url": "https://zenodo.org/records/21793638/files/Paone-Paone_Response-Protection-for-Line-Graph-Equality-Families_v1.0_Zenodo.pdf?download=1",
          "mime_type": "application/pdf",
          "title": "PDF"
        },
        {
          "url": "https://zenodo.org/records/21793638/files/Paone-Paone_Response-Protection_Reproducibility-Package_v1.0_Zenodo_2026-08-04.zip?download=1",
          "mime_type": "application/zip",
          "title": "Zenodo source and reproducibility package"
        }
      ]
    }
  ]
}