Degree Sequences and Poset Structure of Order 9 Graphs

PASS

The argument is a valid counterexample to the conjecture under the natural strict reading of “increase with $n$.” For $n=1$ the multiplicity multiset is $\{1\}$, and for $n=2$ it is $\{1,1\}$, so both median and mean remain $1$. Thus they do not strictly increase from $n=1$ to $n=2$. The proof is elementary and complete for disproving the stated conjunction under that interpretation.

Novelty assessment

KNOWN

Classification rationale: The accepted resolution is only a boundary/wording observation: for $n=1,2$ all degree-sequence multiplicities are $1$, so strict increase of mean/median fails. The underlying small-order enumeration is already standard and tabulated; there is no publishable new combinatorics here.

Literature check: I found no explicit later paper phrasing this as a counterexample to Adams–Eggleton–MacDougall’s sentence. However, stronger known data imply it immediately: OEIS A004251 lists the graphical degree sequences for $n=1,2$ as $(0)$ and $(0,0),(1,1)$; OEIS A122423 gives the number of unigraphic degree sequences on $n$ vertices beginning $1,2,\dots$, so these sequences each have a unique graph realization. OEIS A000088 also gives the corresponding unlabeled graph counts $1,2$. Thus the multiplicity multisets $\{1\}$ and $\{1,1\}$ are already contained in standard enumerative references.

Citation: OEIS A122423, “Number of unigraphic degree sequences among all graphs on $n$ vertices”; OEIS A004251, “Number of graphical partitions”; OEIS A000088, “Number of simple graphs on $n$ unlabeled nodes.” See also S.-Y. R. Li, “Graphic Sequences with Unique Realizations,” JCT B 19 (1975), 42–68.