Some Questions — Question 12

Definitions and mathematical statement

Fix a prime $p$. Consider finite undirected graphs $G_j$ of uniformly bounded degree, with an integer label on each edge and, if needed, an integer diagonal label at each vertex. Let $A_j$ be the corresponding symmetric integer adjacency matrix: an off-diagonal entry is the edge label, or zero for a missing edge, and a diagonal entry is the vertex label.

Root $G_j$ at a uniformly selected vertex. Local convergence means that for every radius $r$, the distribution of the rooted, labeled radius-$r$ neighborhood converges to a probability distribution on such rooted neighborhoods, consistently as $r$ varies. Labels are retained in these distributions, not discarded.

Must the normalized ranks after reducing all entries modulo $p$ converge? \[ \lim_{j\to\infty}\frac{\operatorname{rank}_{\mathbb F_p}(A_j)}{|V(G_j)|} \quad\hbox{exists?} \] Equivalently, ask whether the normalized dimensions of the kernels over the finite field $\mathbb F_p$ converge, since rank plus nullity equals the matrix size.

The matrices are sparse because degrees are bounded, but the dependence among their rows is a global algebraic feature. Real eigenvalue convergence therefore does not by itself prove the asserted finite-field rank convergence. The question includes the ordinary adjacency matrices with every edge label equal to one. The prime is fixed throughout a sequence; varying it with the graph would be a different problem.

Short English statement

Does local convergence of bounded-degree integer-labeled graphs force convergence of adjacency-matrix rank over a fixed finite field?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200012 (AMR-011-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 12. Original source indexed by AMR-011-0012. https://www.renyi.hu/~abert/questions.pdf.