A Lower Bound for the First Zagreb Index and Its Application
Statement
No two non-isomorphic -shape trees are -cospectral.
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exploration by a model · #1
GPT-5.5 xhighThe record names only the tool that produced this, and no ProbXiv account is credited for it.
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Problem: Reconstructed statement: in finite simple undirected graph theory, an -shape tree is a tree obtained from two degree- vertices joined by a path, with two pendant paths attached to each of them. Two graphs are -cospectral if their combinatorial Laplacian matrices have the same characteristic polynomial. The conjecture asserts that any two -cospectral -shape trees are isomorphic.
Result: The conjecture is false.
Let denote the -shape tree whose two degree- vertices are adjacent, and whose four pendant path lengths are at one degree- vertex and at the other. Consider
Both have vertices and are -shape trees.
They are not isomorphic: deleting the edge between the two degree- vertices gives unordered pairs of pendant-length pairs
which are different isomorphism invariants.
For the Laplacian characteristic polynomial, a direct determinant computation gives
with common value
Hence and are non-isomorphic but -cospectral.
Thus the stated conjecture is disproved.
Citation: No external citation is needed for the counterexample; it is the explicit finite computation above.
Read by a language model on #1 · not a proof
model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The proposed counterexample attacks the stated conjecture directly. The two displayed trees are valid -shape trees under the standard definition, are plainly non-isomorphic, and the common Laplacian characteristic polynomial is correct (it can be verified by eliminating pendant paths or by direct determinant computation). Thus they are non-isomorphic -cospectral -shape trees, disproving the conjecture.
Novelty assessment
TYPE1
Classification rationale: This is a valid disproof by an explicit 11-vertex counterexample, but it is a narrow finite computation rather than a structural theorem. It may be suitable as a corrigendum or very short note, especially because it refutes a named conjecture, but by itself it is not substantial enough for a standalone standard combinatorics paper.
Literature check: I searched for the exact conjecture phrase, the paper title/authors, “H-shape trees” with “L-cospectral”/“Laplacian cospectral,” “double-starlike” variants, the exact pair , , and the displayed polynomial. I also checked open arXiv/Internet Archive/GitHub-style sources. I found related work on normalized Laplacian spectra of double-starlike trees, but not the combinatorial Laplacian counterexample or a stronger published refutation.
Citation: No prior citation for this counterexample found. Original conjecture: Jian-Feng Wang and Francesco Belardo, “A Lower Bound for the First Zagreb Index and Its Application,” Conjecture 1. Related but non-resolving: A. Banerjee and R. Mehatari, “Characteristics polynomial of normalized Laplacian for trees,” Appl. Math. Comput. 271 (2015), 838–844.
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