A research direction began in the 1980’s with graph quasi-randomness, extended through the 1990’s and early 2000’s with generalizations to non-uniform graph distributions and other combinatorial objects, became graph limit theory in the mid-2000’s, and culminated in Lovász’s now-canonical text . The central idea is that, if a sequence of graphs with number of vertices tending to infinity has the property that the density of any particular subgraph tends to a limit, then itself tends to a limit object , called a “graphon”. There are several mutually (though non-obviously) equivalent ways to view graphons, and one of them is as a self-adjoint kernel operator from to , an object type for which a well-established spectral theory exists. Therefore, Szegedy () was able to introduce a spectral theory of graphons and show that is a natural analogue of the spectral theory of finite graph adjacency matrices. Here, we extend this perspective by showing that graphons are associated with a power series which is a certain normalized limit of the characteristic polynomial of graphs. Furthermore, we give a straightforward formula for this “characteristic power series” in terms of the edge density and spectrum of the graphon (Theorem 4).
We also show that the characteristic power series can be used to characterize “quasi-randomness”. Suppose that is a sequence of graphs with . (In truth, all that is needed is that , but this is not more general.) We write for simplicity and for the number of labelled, not-necessarily induced copies of as subgraphs in (i.e., homomorphisms from to ). Then, by a classic 1989 paper of Chung, Graham, and Wilson (), there is a large set of random-like properties (properties which hold asymptotically almost surely for graphs in the Erdős-Rényi model ) which are mutually equivalent, and are therefore collectively referred to as (the sequence of graphs) being “quasi-random”. Namely, let
denote the property that the number of labelled occurrences of each graph on vertices as an induced subgraph of is
denote the property that for each graph on vertices.
denotes the property that and , where is the -cycle
denotes the property that , , and , where
are the complete set of adjacency eigenvalues of
denotes the property that, for all , , where denotes the subgraph of induced by
denotes the property that, for all with , , where denotes the subgraph of induced by
denotes the property that
denotes the property that
Theorem 1 (Chung-Graham-Wilson ).
For and even,
If a graph sequence has these properties, it is called -quasi-random, and these properties and any others also equivalent to them are known as (-)quasi-random properties. Many other quasi-random properties have been added since to the list above, such as other families of graphs whose occurrence as subgraphs at the “random-like rate” implies these properties (note that is the “forcing” family given by ), and also that converges to a uniform graphon.
Here, we propose to add another property. First, given a convergent sequence of graphs as above, let be their limit, and let denote the (adjacency) characteristic polynomials of . Define
if the (pointwise) limit exists. We call the “characteristic power series” of the graphon
, and it is essentially a normalized limit of the reciprocal polynomials of the characteristic functions of.
2 Characteristic Power Series of Graphons
The following classical result will be useful in describing the coefficients of .
Theorem 2 (Harary-Sachs ).
The coefficient of in is
where is the number of components of , is the number of cycles of , and is the family of all graphs on vertices each of whose components is an edge or a cycle, and denotes the number of subsets of edges of which are isomorphic to .
Clearly, if , then . Also,
where has components of size for each . For simplicity, for a partition where part occurs times for each (the all distinct), the quantity is defined by
and we write where is the integer partition of given by the component cardinalities of . If is a partition of , then the number of partitions of an -set with structure (a partition with parts of distinct sizes ) is given by
Letting , we have by Theorem 2,
Note that (treating this as a polynomial to avoid defining ), when , the above expression equals because is monic. Denote this polynomial by .
Suppose the graphs converge to the graphon . For the sequence of functions corresponding to the sequence of graphs :
converges pointwise as .
converges uniformly on compact sets as .
Each coefficient of converges as .
Furthermore, the limit is , is entire of Laguerre-Pólya class (locally the limit of a series of polynomials whose roots are all real), and its roots are precisely the reciprocals of the nonzero eigenvalues of (as a self-adjoint kernel operator) with multiplicity.
We may write
Clearly, has finite rank. It converges to in trace class norm, since the spectrum of the former converges to the spectrum of the latter (with multiplicity) in (Lemma 1.8 in ). Thus, for the Fredholm determinant of , Lemma VII.4.1 of  implies that
uniformly on compact sets, and the limit is entire. Then (1) and (2) follow, and, since (2) holds, Cauchy’s integral formula implies that (3) holds as well. Since has only real roots (being the characteristic polynomial of a real symmetric matrix), the limit is of Laguerre-Pólya class. That the roots of are the eigenvalues of the kernel operator corresponding to is an immediate consequence of Lemma VII.6.1 of . ∎
The fact that the limit of is just has immediate consequences from various properties of Fredholm determinants, e.g.:
Given two graphons and , the graphon which is their disjoint union has the property that
This follows from the fact that the Fredholm determinant satisfies . ∎
The property of a sequence of graphs that converges pointwise to a function with only one root (of multiplicity one) at is a -quasi-random property.
Let be the -uniform graphon, i.e., the limit of a -quasirandom graph sequence. Convergence of the coefficients of to those of , which is implied by pointwise convergence by Cauchy’s Integral Formula, is equivalent to -quasirandomness, by property . Therefore, by Theorem 1, converges pointwise to if and only if converges to , i.e., is -quasirandom. By Lemma 1 and Hurwitz’s Theorem, coefficient convergence implies root convergence, in the sense that every ball, for sufficiently small, about a zero of of multiplicity will contain exactly roots of for sufficiently large , including for roots at infinity. If is -quasirandom, then by the eigenvalues of are (once) and (with multiplicity ), so the roots of are (once) and some roots the smallest modulus of which tends to infinity. Thus, has a root of multiplicity one at and no other roots. ∎
One consequence of the above result is that the set of linear combinations of subgraph counts (a.k.a. the density statistics of “quantum graphs”) occurring as the coefficients of powers of in are forcing sets for -quasirandomness. Actually, this is easy to see directly. If the edge density is , then the coefficient of is
and the coefficient of is
Therefore, the coefficients of and are -random-like iff the quantity is -random-like (i.e., ), iff all of the coefficients of are -random-like because of in Theorem 1 above.
The fact that is Laguerre-Pólya class implies that it has a Hadamard product expression (see ):
where is a nonnegative integer; and are real with ; and ranges over the nonzero zeros of . We can use this to obtain a formula for the characteristic polynomial of any graphon as a function of its nonzero eigenvalues and its edge density, since the roots of are the reciprocals of the nonzero eigenvalues of .
Letting be the edge-density of a graphon (i.e., ) and its spectrum, we have
First, observe that
implies that and because , since the constant coefficient of is . Furthermore, we may include the zero eigenvalues in this product without effect. Then
whose term will be , since has zero linear term. But has zero linear term, so , and
Now, the coefficient of in this expression is
but will also be , where is the edge density of , as per (1). Thus, , so . (The sum converges because Laguerre-Pólya class functions always have convergent sum of the reciprocals of their roots; this is also a consequence of .) ∎
In the expression
the characteristic power series has been factored into a polynomial whose roots are the reciprocals of the nonzero eigenvalues of and an exponential term. The quantity is zero if is bipartite, in which case
has no monomials of odd degree:
Furthermore, iff is a - function except for a set of measure zero, as with a simple blow-up of a graph (sometimes called a “pixel diagram”), so the quadratic term vanishes in the exponential, resulting in
We can also use (2) to obtain a simple expression for the characteristic power series of -quasi-random graphons.
is -quasirandom iff
where is the set of integer partitions of into parts of size at least , of which are of size exactly .
By Theorem 3, is -quasirandom iff
because partitions of into parts of size at least correspond bijectively to elements of . That this equals the right-hand side of (3) follows from Theorem 4 and the fact (from, say, , section 11.8, or the proof of 3 above) that a -quasirandom graphon has a single nonzero eigenvalue of multiplicity one. However, we show the result directly here.
If is a power series, where the coefficient of is and the coefficient of is , then (by standard facts about exponential generating functions, see, e.g., ), letting be an integer partition with parts of distinct sizes , to ,
where is the set of (set) partitions of an -set. Thus, if
and , then
Here we leave the interested reader with a few questions.
For which other graph polynomials can the above type of limit process be carried out? By , chromatic polynomials of Erdős-Rényi random graphs appear to have a scaling limit, for example.
It is straightforward to show (by, for example, applying Turán’s Inequalities; see ) that the coefficients of are log-concave for any graphon , but the consequences of this for unimodality are unclear because we do not know the sign pattern of the coefficients of . For example, the characteristic power series of a quasi-random graphon has sign pattern
More specifically, can if ?
How can the above analysis be extended to sparse graphon types or hypergraphs? Perhaps, one can use the below generalization of Theorem 2 to hypergraphs.
Theorem 5 (Clark-Cooper ).
Let be a -uniform hypergraph on vertices. If denotes the set of -uniform Veblen multi-hypergraphs (i.e., all vertices have degree divisible by ), and denotes the codegree- coefficient in the characteristic polynomial of , then
where is the number of components of , is a particular easily-computable function of , and is the number of a certain type of structure-preserving map of to subgraphs of .
Thanks to Greg Clark, Chris Edgar, Vlado Nikiforov, and Alex Riasanovsky for helpful discussions.
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