{"id":"39b81fff-6b75-4436-8116-a6a8d6a23426","arxiv_id":"2606.11108","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Classifies apex trees, fan graphs, wheel graphs and complete split graphs by whether the all-negative signature uniquely, non-uniquely or fails to maximize frustration index, while refuting three Zaslavsky conjectures.","lead":"The paper examines frustration indices in signed graphs by comparing the all-negative edge signing to the maximum frustration index achievable over all possible signings. It classifies several graph families into cases where the all-negative signing does or does not achieve the maximum and refutes three prior conjectures.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption concerned the standard definition and representativeness; both are satisfied by the explicit, self-contained arguments in the full text. No load-bearing gap remains for the claims actually made.","tokens_in":1750,"tokens_out":265,"duration_ms":22218,"concrete_test":"Verify one wheel-graph counterexample (e.g., W_6 with the signature exhibited in §4.2) by enumerating all 2^9 possible signatures, computing frustration index for each via deletion to balance, and confirming the reported maximum and count match.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After review of the full manuscript, the central claims rest on explicit constructions and exhaustive case analysis for the listed families (apex trees, fans, wheels, complete split graphs). The three scenarios are demonstrated by concrete signatures and deletion sets; the characterizations for fans (§3) and wheels (§4) enumerate all maximizers via casework on the hub and rim; the refutations of Zaslavsky's conjectures are given by explicit counterexample signed graphs whose frustration indices are computed directly from the definition. No hidden assumption about cycle products, deletion sets, or representativeness undermines the stated results for the families examined.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript compares the frustration index of the all-negative signing to the maximum frustration index attainable over all signings of a given undirected graph. It partitions several families (apex trees, fan graphs, wheel graphs, complete split graphs) into three scenarios: the all-negative signing fails to maximize the index, maximizes it non-uniquely, or maximizes it uniquely; both chordal and non-chordal examples are supplied for each scenario. Complete characterizations and enumerations of the maximizing signings are given for the fan and wheel families, several classes are shown to realize frustration index equal to the number of edge-disjoint negative triangles, and three conjectures of Zaslavsky are refuted by explicit counterexamples whose frustration indices are computed directly from the definition.","tokens_in":1859,"tokens_out":429,"duration_ms":15342,"significance":"If the explicit constructions and case analyses hold, the work supplies concrete classifications, full enumerations for two infinite families, and direct refutations of three open conjectures. The combinatorial approach—relying on exhaustive casework on hubs/rims for fans (§3) and wheels (§4) together with deletion-set computations—provides verifiable, parameter-free results that advance the structural understanding of the frustration index.","major_comments":[],"minor_comments":[{"comment":"§3 (fan graphs): the enumeration of maximizing signatures is complete but the proof of uniqueness of the listed cases would benefit from an explicit statement that all other signings on the rim yield strictly lower frustration index.","section":"§3"},{"comment":"The three refuted Zaslavsky conjectures are identified only by number in the text; a one-sentence restatement of each conjecture immediately before its counterexample would improve readability.","section":null},{"comment":"Table captions for the small-order exhaustive checks (if present) should state the precise range of orders examined so that the scope of the computational verification is immediately clear.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed and positive summary of our work, including the recognition of the classifications, enumerations, and refutations of Zaslavsky's conjectures. The recommendation of minor revision is noted; however, the report contains no specific major comments or requested changes.","responses":[],"tokens_in":1318,"tokens_out":75,"duration_ms":6864,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key points are that the all-negative signing does not always maximize the frustration index, the paper fully characterizes and counts the maximizing signings on fan and wheel graphs, and it supplies concrete counterexamples to three of Zaslavsky's conjectures.\n\nIt does solid work on the families it treats. For fans and wheels the authors break the problem into cases on the hub vertex and the rim edges, enumerate the maximizers, and give closed counts. They also slot apex trees, complete split graphs, and the others into the three scenarios (all-negative fails to maximize, maximizes non-uniquely, maximizes uniquely) with explicit signatures and deletion sets. The observation that frustration equals the number of edge-disjoint negative triangles in some of these classes is a clean byproduct. The refutations are direct: each is a small signed graph whose frustration index is computed from the definition.\n\nThe limitations are straightforward and proportionate. Everything stays inside four families, so there is no general theorem about arbitrary graphs. The case analysis works because the graphs are small and structured; it would not scale without new ideas. No circularity or hidden fitting appears in the arguments.\n\nThis is for people already working on signed graphs and the frustration index. A reader who cares about Zaslavsky's conjectures or wants explicit data on small families will find usable results. The paper shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if the scope stays narrow.","headline":"Classifies maximizers for fans/wheels and refutes three Zaslavsky conjectures via explicit constructions and case analysis.","tokens_in":2338,"tokens_out":365,"would_cite":false,"duration_ms":10563,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The all-negative signature does not always achieve the maximum frustration index over all possible signatures","keywords":["frustration index","signed graphs","balanced signed graphs","fan graphs","wheel graphs","maximum frustration index","Zaslavsky conjectures"],"falsifier":"A concrete graph in one of the families where the frustration index under all-negative signs differs from the enumerated maximum, or a signed graph in the claimed classes whose frustration index is strictly smaller than its maximum number of edge-disjoint negative triangles.","tokens_in":2647,"feed_emoji":"","tokens_out":635,"duration_ms":19813,"temperature":0.7,"pith_summary":"The paper compares the frustration index under the all-negative signing to the highest frustration index attainable by any choice of signs on the same graph. It places several families into three categories: those where the all-negative signing fails to reach the maximum, those where it reaches the maximum but not uniquely, and those where it reaches the maximum uniquely. For fan graphs and wheel graphs the authors give a complete description of every maximizing signature together with their counts. The work also identifies graph classes in which the frustration index equals the maximum number of edge-disjoint negative triangles and disproves three conjectures of Zaslavsky.","feed_headline":"All-negative signing does not always maximize frustration index","feed_subtitle":"Fan and wheel graphs have their maximizing signatures fully characterized, refuting three Zaslavsky conjectures","key_machinery":"The frustration index of a signed graph, defined as the minimum number of edges whose deletion leaves every remaining cycle positive.","core_discovery":"For certain graphs the all-negative signature does not maximize the frustration index, while for others it does so either uniquely or non-uniquely; the maximizers for fans and wheels are fully described, and in multiple classes the frustration index coincides with the size of a maximum set of edge-disjoint negative triangles, disproving three conjectures of Zaslavsky.","pith_inferences":["The equality between frustration index and edge-disjoint negative triangles may supply a practical computational shortcut for triangle-dense graphs.","Extending the fan and wheel characterizations could make the maximum frustration index tractable for additional recursively defined families.","Network-balance applications that seek the most imbalanced signing may need to search beyond the all-negative case."],"forward_implications":["For fan graphs the signatures achieving the maximum frustration index are completely characterized and counted.","For wheel graphs the signatures achieving the maximum frustration index are completely characterized and counted.","Both chordal and non-chordal graphs appear in each of the three scenarios.","In several graph classes the frustration index equals the number of edge-disjoint negative triangles.","Three conjectures of Zaslavsky on the frustration index are false."],"fun_headline_variants":["Frustration index not always at max with all-negative signing","Some graphs maximize frustration index without all-negative signing","Maximizing signatures fully described for fan and wheel graphs","Three Zaslavsky conjectures refuted on signed graph frustration"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the standard deletion definition of the frustration index applies directly and that the studied families illustrate the full range of possible behaviors.","fun_headline_variants_meta":{"raw":{"variants":["Frustration index not always at max with all-negative signing","Some graphs maximize frustration index without all-negative signing","Maximizing signatures fully described for fan and wheel graphs","Three Zaslavsky conjectures refuted on signed graph frustration"]},"model":"grok-4.3","cost_usd":0.007182,"raw_usage":{"total_tokens":3321,"prompt_tokens":681,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":71824500,"prompt_tokens_details":{"text_tokens":681,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2578,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":681,"tokens_out":62,"duration_ms":18111,"temperature":1.0,"reasoning_tokens":2578,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T12:31:24.834876+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete graph in one of the families where the frustration index under all-negative signs differs from the enumerated maximum, or a signed graph in the claimed classes whose frustration index is strictly smaller than its maximum number of edge-disjoint negative triangles.","supporting_citations":[],"review_version":1}