{"id":"ede62509-ea8d-4422-8c65-a2d7646002ee","arxiv_id":"1908.01001","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims the automorphism group of a nonzero component graph is the symmetric group on the basis and derives distinguishing numbers, but the automorphism claim fails for q≥3 and the q=2 proof has a gap.","lead":"The paper computes distinguishing numbers for graphs built from nonzero vectors of finite vector spaces. A central claim about the automorphism group is false for fields with more than two elements, and the proof for the binary-field case contains a concrete error.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.10 fails for q>=3: monomial maps b_i -> c_i b_{sigma(i)} are automorphisms but do not permute the chosen basis, so Theorem 2.11's isomorphism claim is false.","rationale":"The reader's strongest claim correctly identifies the central false result: Sym is not isomorphic to Aut(G(V)) for q>=3. The paper's Corollary 2.10, which is the bridge between arbitrary automorphisms and permutations of the chosen basis, fails exactly because automorphisms may send a basis vector to a nonzero scalar multiple of another basis vector. Such maps are graph automorphisms whenever the scalar is nonzero, since they simply permute coordinates and scale them, preserving whether two supports intersect. The proof of Theorem 2.11 relies entirely on Corollary 2.10 for surjectivity, so the isomorphism proof collapses. The abstract states this isomorphism without the q=2 restriction, so the paper's headline claim is false as written. Additionally, Theorem 2.12's cardinality claim n! is false for q>=3. These are not cosmetic issues: the distinguishing-number results for q>=3 in Theorem 3.4 are couched in terms of an automorphism group that the paper has not correctly characterized. Even for q=2, the proof of Theorem 3.2 has gaps: Lemma 3.1's inference from f(u) != f(v) to the breaking of transpositions is not justified because an automorphism swapping bl and bm need not map u to v, and the T_{n-1} label sets in the proof are ill-defined for small n. Thus rejection is warranted.","tokens_in":8440,"tokens_out":9484,"duration_ms":103604,"concrete_test":"Compute the automorphism group of G(F_3^3) by brute force (or by a short script): generate the 26 non-zero vectors, define adjacency as nonempty intersection of coordinate supports, and count automorphisms. Verify that phi(x1,x2,x3)=(2x1,x2,x3) is an automorphism. Since phi(e1)=2e1 is not in {e1,e2,e3}, this automorphism is not induced by any permutation of the basis, and the total number of automorphisms is at least 48, not 6. This directly refutes Theorems 2.11 and 2.12 for q=3.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing assumption is Corollary 2.10, used in the onto step of Theorem 2.11 and in Theorem 2.12. It asserts that every automorphism restricts to a permutation of the fixed basis {b1,...,bn}. This is false for q>=3. For any permutation sigma and any nonzero field elements c_i, define phi(sum alpha_i b_i) = sum c_i alpha_i b_{sigma(i)}. Since all c_i are nonzero, the support of a vector is transformed exactly as support(phi(x)) = sigma(support(x)); hence supports intersect if and only if their images intersect. Therefore phi is a graph automorphism of G(V). For a concrete case, take V=F_3^3 and phi(x1,x2,x3)=(2x1,x2,x3). This nontrivial automorphism fixes b2 and b3 but sends b1 to 2b1, which is not one of the chosen basis vectors. Thus the restriction of phi to T1 is not a permutation of {b1,b2,b3}, contradicting Corollary 2.10. Consequently, the proof of Theorem 2.11 cannot produce phi from any element of Sym, and the isomorphism Sym ~= Aut(G(V)) is false. In fact Aut(G(V)) contains the full monomial group (q-1)^n semidirect S_n, so |Aut(G(V))| is (q-1)^n n!, not n!, for q>=3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the non-zero component graph G(V) associated with a finite-dimensional vector space V over a field with q elements. The main claims are: (1) every permutation of a fixed basis extends to an automorphism of G(V); (2) the symmetric group on the basis vectors is isomorphic to Aut(G(V)); (3) consequently |Aut(G(V))|=n!; (4) the distinguishing number is 2 when q=2 and n≥3; and (5) the distinguishing number is (q−1)^n when q≥3 and n≥3. The abstract states the isomorphism and cardinality results without any restriction on q, while Section 2 begins by restricting attention to the field with two elements. Section 3 treats the q=2 and q≥3 cases separately, using a labeling argument for q=2 and a twin-set argument for q≥3.","tokens_in":8743,"tokens_out":14582,"duration_ms":138333,"significance":"If fully established, the results would provide a complete description of the automorphism group and the distinguishing number of non-zero component graphs, a natural and useful contribution to the study of graphs associated to vector spaces. The paper does not include machine-checked proofs or reproducible code; its contribution is purely mathematical. Some parts are likely correct, in particular the q=2 automorphism result (which follows from earlier work of Das and of the authors' group) and the lower bound in the q≥3 distinguishing result. However, the manuscript as written contains false unqualified statements and a central proof (Theorem 3.2) that relies on a false lemma, so the claimed results are not supported as stated.","major_comments":[{"comment":"The isomorphism Sym ≅ Aut(G(V)) and the consequence |Aut(G(V))|=n! are false when q≥3. For example, over F_3 with n=3, the map φ(x1,x2,x3)=(2x1,x2,x3) is an automorphism of G(V): it is bijective on nonzero vectors and preserves skeletons, hence preserves adjacency and non-adjacency. This automorphism sends b1 to 2b1, which is not one of the chosen basis vectors, so it is not induced by a permutation of {b1,b2,b3}. The monomial group (F_q^×)^n ⋊ S_n is contained in Aut(G(V)) for q≥3. The introduction to Section 2 explicitly restricts to fields with two elements, but the abstract and the statements of Theorem 2.11 and Theorem 2.12 do not carry this restriction. This is a load-bearing overgeneralization that must be corrected throughout the paper.","section":"Abstract and §2, Theorem 2.11 and Theorem 2.12"},{"comment":"The extension construction in Theorem 2.9 is not well-defined for q≥3. The proof defines g(u) as 'a vector whose skeleton is {h(b1),...,h(bm)}'; when q>2 there are (q−1)^m distinct vectors with that skeleton, so the image of u is not uniquely specified. The construction works only when q=2, where each skeleton determines a unique vector. The theorem statement and its proof require the hypothesis q=2 (or an explicit rule for transforming the coefficients).","section":"§2, Theorem 2.9"},{"comment":"Lemma 3.1 is false as stated. Take n=6, q=2, u=b1+b3+b5, v=b2+b3+b6, and let g be the automorphism that swaps b1 and b2 and fixes all other basis vectors. Then b1∈S_u\\S_v and b2∈S_v\\S_u. Define a 2-labeling f by f(u)=f(g(u))=1, f(v)=f(g(v))=2, and f(w)=1 for all other vertices. Then f(u)≠f(v), but g preserves f, so f does not break g even though g maps b1 and b2 to each other. The deduction in the proof, that because u and v cannot map to each other Lemma 2.6(i) prevents b_l and b_m from mapping to each other, is invalid: an automorphism swapping b_l and b_m need not send u to v.","section":"§3, Lemma 3.1"},{"comment":"The labeling of T_{n−1} is incorrect. The sets S_u={b2,b3,...,b_{⌊n/2⌋}} and S_v={b1,b3,b4,...,b_{⌊n/2⌋}} have size ⌊n/2⌋−1 and ⌊n/2⌋−1 respectively, not n−1, so these vertices do not belong to T_{n−1} for n≥5. Thus the proposed labels do not apply to the intended class, and the claimed destruction of the transposition (b1 b2) is not established.","section":"§3, Theorem 3.2, step (2)"},{"comment":"The proof of Theorem 3.4 is incomplete. The statement that each twin set T_i^k is labeled with (q−1)^i labels 'out of' the (q−1)^n available labels, and that this destroys all automorphisms of disjoint twin sets, does not address automorphisms that map one twin set to another. If two twin sets receive the same set of labels, a nontrivial automorphism could map one to the other while preserving labels. The result is likely true — an injective labeling of T_n with (q−1)^n colors would break every automorphism — but the argument as written needs to be replaced by a rigorous global construction.","section":"§3, Theorem 3.4"}],"minor_comments":[{"comment":"The phrase 'for both of the cases' should read 'in both cases'.","section":"Abstract"},{"comment":"There is a grammatical error: 'properties of automorphisms non-zero component graphs' should be 'properties of automorphisms of non-zero component graphs'.","section":"§1, last sentence of introduction"},{"comment":"The statement reads 'Let GV be the non-zero component graph'; it should be 'Let G(V) be the non-zero component graph'.","section":"§3, Theorem 3.4 statement"},{"comment":"The paper repeatedly refers to 'we' in describing results from references [5] and [15] (e.g., Lemmas 2.2–2.6 and the partition into classes T_i). These should be attributed in the third person, e.g., 'it was shown in [5]'.","section":"Throughout"},{"comment":"In the well-definedness part (1), the proof assumes that for a vertex u∈T_m the skeleton is S_u={b1,...,bm}. This is only true after relabeling the basis; the proof should say 'without loss of generality'.","section":"§2, proof of Theorem 2.11"},{"comment":"Reference [12] contains the typo 'functi-graphs'; it should be 'function graphs'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's Section 2 is explicitly confined to q=2, so the automorphism isomorphism is not false in context; however, the abstract and the theorem statements overclaim without the q=2 hypothesis, and this must be fixed. More seriously, the proof of the q=2 distinguishing number (Theorem 3.2) relies on the false Lemma 3.1 and contains an incorrect T_{n−1} labeling. Since the result itself is plausibly correct, a substantial revision with new proofs for Section 3 could make the paper publishable. The q≥3 distinguishing-number result also needs a rigorous proof, though its statement appears correct. Given the number of load-bearing gaps, I recommend major revision rather than acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nI read the paper and the reader's report. I agree with the reject verdict, and the stress-test note is right on target. The central claim—Theorem 2.11 and the abstract's unqualified statement that the symmetric group of basis vectors is isomorphic to Aut(G(V))—is false for q≥3. A monomial map sending each basis vector to a nonzero scalar multiple of a permuted basis vector preserves support intersections, so it is an automorphism; for q≥3 these maps are not just permutations of the basis. For V=F_3^3, (x1,x2,x3) ↦ (2x1,x2,x3) fixes b2,b3 and sends b1 to 2b1, so it does not restrict to a permutation of the chosen basis. Corollary 2.10, used to prove onto-ness in Theorem 2.11, fails. The automorphism group is actually the monomial group of order (q-1)^n n!, so Theorem 2.12 is also off by a factor of (q-1)^n.\n\nWhat is worth keeping: Theorem 2.9 correctly shows every basis permutation extends to an automorphism. For q=2, scalar multiples collapse, so the symmetric-group isomorphism and |Aut|=n! are valid. The distinguishing-number results are the new part, and the q=2 result Dist=2 is plausible, but the proof is not sound. Lemma 3.1 is invalid: two vertices with different labels being unable to map to each other does not break an automorphism that sends one of them to a third vertex with the same label. The verification in the T_{n-1} step of Theorem 3.2 also uses sets S_u and S_v that do not have size n-1. For q≥3, Dist=(q-1)^n is actually the right answer, but the paper's twin-set labeling proof is incomplete; the clean argument is that labeling the largest twin set (all full-support vectors) injectively is already distinguishing, since the monomial action is faithful on that set.\n\nThe paper leans heavily on self-citations [5] and [15] for its key lemmas, but the main failure is mathematical, not bibliographic. The intended audience is researchers working on symmetry breaking in graphs from vector spaces; a corrected version could be a minor contribution. As it stands, the main theorem is false and the q=2 proof has real gaps. I wouldn't send this to peer review; a desk reject is justified, with an invitation to resubmit a corrected version.","headline":"The paper's central isomorphism claim is false for q≥3, and the q=2 distinguishing proof has real gaps; the salvageable kernel is small.","tokens_in":9196,"tokens_out":10427,"would_cite":false,"duration_ms":103538,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:53:26.347355+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}