Quasi-random graphs, subgraph counts and graph limits, again

Abstract: We study properties of graphs (or rather graph sequences) saying that some restricted count of subgraphs is approximatively what is expected in a random graph. It has been shown by several authors that many such properties characterize quasi-random graphs, but there are also some exceptions. We continue here the line of investigation in Janson and Sós (2013), and introduce some new versions of these properties, in order to better understand why many of these properties are quasi-random, and to understand the structure of the exceptions that are not. A new feature in the proofs is a simple decomposition of the subspace of symmetric functions in $L^2([0,1]^m)$ into subspaces that are irreducible for the action of measure-preserving transformations of $[0,1]$; this simplifies some arguments and gives structure to others.

• The paper provides a structural characterization of quasi-randomness by linking subgraph count properties to graph limits and constant graphon conditions.
• It introduces an irreducible decomposition of symmetric function spaces using group representation theory to pinpoint exceptions in quasi-random behavior.
• The findings enable rigorous testing of large networks for quasi-random properties and offer algebraic criteria to identify rare non-quasi-random cases.

Quasi-random Graphs, Subgraph Counts, and Graph Limits: Structural Characterization and Irreducible Decomposition

Introduction and Context

The paper "Quasi-random graphs, subgraph counts and graph limits, again" (2606.20186) presents a rigorous investigation of asymptotic graph properties defined via subgraph counts, their relationship to quasi-randomness, and their translation in the framework of graph limits (graphons). The work refines and extends previous characterizations of quasi-randomness, building on foundational results by Chung, Graham, Wilson, Thomason, Lovász, and Szegedy. It addresses cases where restricted subgraph count properties are (or are not) sufficient to characterize quasi-random graph sequences and provides a structural decomposition for the function spaces underlying these properties, exploiting group representation theory.

Subgraph Count Properties and Quasi-randomness

The paper considers graph sequences $(G_n)$ with $|G_n| \to \infty$ and defines quasi-randomness through several equivalent "random-like" properties (cuts, edge distributions, etc.). A central question is: which restricted subgraph count properties imply quasi-randomness? Specifically, the focus is on counts of labeled or unlabeled copies of a fixed graph $F$ within subsets $U_1,\dots,U_r$ of $V(G)$, parameterized by $\mathbf{m} = (m_1,\dots,m_r)$ and subset proportions $\boldsymbol{\alpha} = (\alpha_1,\dots,\alpha_r)$. These counts are further symmetrized to eliminate labeling-dependent anomalies, yielding properties such as $(F; \mathbf{m}, \boldsymbol{\alpha})$, which state that the relevant subgraph count matches $p^{e(F)} \prod_i |U_i|^{m_i} + o(|G_n|^m)$ for all admissible subset choices.

It is established that many such properties are quasi-random for generic parameters but admit exceptional cases, notably for certain balanced cuts (e.g., counts for $K_2$ over subsets of size $|G_n| \to \infty$0 are not sufficient), and for full-vertex-induced counts.

Graphon Translation and Functional Analytic Framework

The analytic translation uses the graphon framework: a symmetric measurable function $|G_n| \to \infty$1 encodes edge probabilities in the limit. Subgraph count properties translate to integral equations involving the graphon and symmetric subsets $|G_n| \to \infty$2: $|G_n| \to \infty$3 for all disjoint measurable $|G_n| \to \infty$4 with $|G_n| \to \infty$5.

The paper shows that verifying quasi-randomness for graph sequences is equivalent to requiring that only constant graphons satisfy these equations.

Irreducible Decomposition via Representation Theory

A key technical advance is a decomposition of the Hilbert space $|G_n| \to \infty$6 (the $|G_n| \to \infty$7-variable function space of subgraph types) into irreducible subspaces for the action of measure-preserving bijections of $|G_n| \to \infty$8. This is achieved by constructing spaces of symmetric functions dependent only on $|G_n| \to \infty$9 variables, denoted $F$0, and establishing their irreducibility and non-equivalence.

The main theorem is that every closed invariant subspace is a direct sum of these irreducible spaces: $F$1 for some $F$2. Consequently, the analysis of quasi-random properties is reduced to examining which $F$3 (called "bad" indices) are present in the kernel of the associated operator.

Algebraic Characterization of Exceptional Cases

For generic parameter values $F$4, all irreducible components except $F$5 are absent ("no bad $F$6"), and the subgraph count property yields quasi-randomness. Strong results are proven for all $F$7, and for many other configurations.

The exceptional cases are characterized algebraically. Notably, when $F$8 (balanced partitions) and $F$9, the property fails. These exceptions correlate with when certain elementary symmetric polynomials vanish for all vector choices summing to zero, yielding divisibility conditions and polynomial identities involving subgraph counts.

The existence of non-constant graphons satisfying these identities translates to open algebraic conjectures about possible parameter values, particularly for $U_1,\dots,U_r$0 beyond $U_1,\dots,U_r$1 or non-regular graphs.

Numerical and Structural Results

• Generic Quasi-randomness: For almost all parameter choices, the space of symmetric functions satisfying the subgraph count property is trivial, confirming quasi-randomness. Only finitely many parameter vectors allow non-trivial kernels, thus exceptional cases are structurally isolated.
• Irreducible Structure: The decomposition provides a complete description of invariant subspaces for measure-preserving transformations. Each $U_1,\dots,U_r$2 corresponds to functions depending only on $U_1,\dots,U_r$3 variables, and the criterion for a bad $U_1,\dots,U_r$4 is given in polynomial algebra.
• Algebraic Reductions: For specific graphs and partition parameters, the characterization boils down to divisibility and zero conditions for symmetric polynomials, allowing algorithmic verification of quasi-randomness or identification of exceptions.
• Known Counterexamples: Only the classic $U_1,\dots,U_r$5 balanced cut, full-vertex counts, and certain graphs with isolated vertices are excluded from the quasi-random regime.
• Extension to Induced Counts and Nonsymmetric Versions: The paper discusses extensions to induced subgraph counts and nonsymmetric subgraph count properties, noting that these are likely more complex but amenable to similar decomposition in principle.

Implications and Open Questions

The structural irreducible decomposition profoundly clarifies which graphon properties are quasi-random and precisely delineates the nature and rarity of exceptions. The connection to group representation theory indicates a robust framework for related combinatorial and analytic problems.

Practically, these results enable rigorous testing for quasi-randomness from subgraph statistics in large networks and provide theoretical underpinning for property testing and graph sampling algorithms.

Open algebraic conjectures remain for the characterization of all exceptional cases, especially for higher-order partitions and complex graphs. The problem of whether 2-type graphons suffice to produce counterexamples in all non-quasi-random regimes remains unsolved.

Conclusion

This paper establishes a comprehensive analytic and algebraic framework for subgraph count-based quasi-randomness in graphs and their limits. The introduction of an irreducible decomposition within symmetric function spaces clarifies the structure of properties satisfied by random-like graphs and translates the characterization problem into algebraic conditions on symmetric polynomials. The findings confirm that quasi-randomness is generic for subgraph count properties, with explicit and well-understood exceptions. The work sets the stage for further exploration of induced counts, nonsymmetrical partitions, and algebraic classification of exceptional cases (2606.20186).