Adjacency matrix equations and related problems: Research notes
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Statement
We conjecture that if a -degree sequence has a tree realization then each such a realization of has the same number of pendant vertices.
Context
Candidate 1 of the open problems stated in "Adjacency matrix equations and related problems: Research notes", extracted for the Scalable Mathematical Discovery run.
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NEW
Problem: For a finite simple undirected graph , define
The -degree sequence is the nonincreasing list of the . The conjecture says: if a -degree sequence has a tree realization, then all its tree realizations have the same number of pendant vertices.
Result: The conjecture is false. Let
On vertices , define by edges
It is connected with vertices and edges, hence a tree. Its degree list by vertex is
so it has pendant vertices. Its -degrees are
which rearrange to .
Define by edges
Again this is a tree. Its degree list is
so it has pendant vertices. Its -degrees are
also rearranging to .
Thus the same -degree sequence has two tree realizations with different numbers of pendant vertices.
Citation: No resolving published citation known. Original conjecture: Maciej M. Sysło, “Adjacency matrix equations and related problems: Research notes,” §4.
Reviews
0 human reviews · 1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification TYPE1
PASS
The counterexample is valid. Both listed graphs are simple connected 12-vertex, 11-edge trees. The computed neighbor-degree sums sort to the same -degree sequence
but has 6 pendant vertices while has 5. This directly disproves the conjecture as stated.
Novelty assessment
TYPE1
Classification rationale: The accepted result is a valid explicit counterexample, but it is a small witness to a low-visibility conjecture, with no accompanying structural theorem, minimality proof, or general construction. If new, it would at most justify a short note/comment, not a standalone standard combinatorics paper.
Literature check: I found no prior published counterexample or stronger known result. Searches covered the original Sysło paper and citation neighborhood, exact/variant terms such as “Gamma-degree sequence,” “Γ-degree sequence,” “Gamma-graphical sequences,” “2-degree sequence,” “neighborhood degree sequence,” “sum of neighbor degrees,” “tree realization,” “pendant vertices,” and the exact sequence . Related work on neighborhood-degree lists and prescribed neighbor-degree sums does not appear to resolve this unpaired -degree-sequence conjecture.
Citation: Maciej M. Sysło, “Adjacency matrix equations and related problems: Research notes,” Commentationes Mathematicae Universitatis Carolinae 24 (1983), no. 2, 211–222. Original conjecture; no prior resolution found.
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