Graph Complexity and Mahler Measure
The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When is -periodic (i.e., has a free -action by graph automorphisms with finite quotient) the Mahler measure of its Laplacian determinant polynomial is the growth rate of the complexity of finite quotients of . Lehmer’s question, an open question about the roots of monic integral polynomials, is equivalent to a question about the complexity growth of edge-weighted 1-periodic graphs.
MSC: 05C10, 37B10, 57M25, 82B20
The complexity of a finite graph is often defined as the number of its spanning trees. Here we consider graphs with integer edge weights and take a different approach, defining complexity to be the order of the torsion subgroup of the abelian group presented by the Laplacian matrix of the graph. When is connected and all edge weights are , the complexity as we define it is the number of spanning trees of the graph. However, for general edge-weighted graphs, the two notions of complexity are different.
Our main objects of study are -periodic graphs, infinite graphs on which , acts freely by graph automorphisms such that integer edge weights are preserved and the quotient graph is finite. Our motivation comes from two sources: knot theory, where finite graphs with edge weights correspond to diagrams of knots and links, and Lehmer’s question concerning the roots of integral polynomials.
For -periodic graphs a Laplacian operator is defined (see ), described by a matrix with variables and denoted here by . We call its determinant the Laplacian (determinant) polynomial . (For finite graphs the term Laplacian polynomial is often used for the characteristic polynomial of the integer Laplacian matrix.) When , we use the main result of  to characterize the Mahler measure of as the complexity growth rate of the finite quotient graphs lying between and . When all edge weights of are , we recover a consequence of a more general result of Lyons  (see also ). We show that Lehmer’s question is equivalent to a question about graph complexity growth rates of 1-periodic graphs with edge weights equal to .
A 1-periodic plane graph (that is, a graph embedded in the plane) determines an infinite link by the medial construction. Its quotient by the -action can be regarded as a (finite) link in an unknotted thickened annulus. The Alexander polynomial of the complement determines the Laplacian polynomial of . We follow with some speculations about Lehmer’s question and links.
In the last section we present useful results about unweighted -periodic graphs. They will not surprise some experts. However, as far as we know they do not appear in previous literature.
Acknowledgements. It is the authors’ pleasure to thank Abhijit Champanerkar, Eriko Hironaka, Matilde Lalin and Chris Smythe for helpful comments and suggestions.
2 Complexity of finite graphs.
Consider a finite graph with vertex set and edge set . The graph is allowed to have multiple edges. Loops will not affect affect results here and can be ignored. We assume also that the edges have weights . (Generally will be or .) The graph is unweighted if every weight is .
The adjacency matrix of is the matrix such that is the sum of the weights of edges between and , for . Diagonal entries of are zero. Define to be the diagonal matrix with
The Laplacian matrix of a finite graph is . The abelian group presented by is the Laplacian group of , denoted by . The (torsion) complexity is the order of the torsion subgroup
When is connected and unweighted, the nullity of the Laplacian matrix is equal to 1 (see ), and hence the Laplacian group decomposes as the direct sum of and the torsion subgroup . In this case, the Matrix Tree Theorem  implies that is equal to the number of spanning trees of .
More generally we define tree complexity of a connected graph by
where the summation is taken over all spanning trees of . If not connected, then we define to be the product of the tree complexities of its connected components. Again by , we have if and only if is nonzero; for connected this common value is equal to any principal minor of . However, the following example shows that can vanish while is positive.
Consider the connected graph in Figure 1. Unlabeled edges here and throughout will be assumed to have weight . The Laplacian matrix is square of size . (See Example 3.8 below for a quick way to find .) A routine calculation shows that any principal minor of vanishes, and hence . However, the absolute value of the greatest common divisors of the minors of is 9. Hence .
3 -Periodic graphs.
We regard as the multiplicative abelian group freely generated by . We denote the Laurent polynomial ring by . As an abelian group is generated freely by monomials , where .
Let be a graph that is -periodic. By this we mean that has a cofinite free -action by automorphisms that preserves edge weights. (By cofinite we mean the quotient graph is finite. The action is free if the stabilizer of any edge or vertex is trivial.) Such a graph is necessarily locally finite. The vertex set and the edge set consist of finitely many vertex orbits and weighted-edge orbits , respectively. The -action is determined by
where and . (When is embedded in some Euclidean space with acting by translation, it is usually called a lattice graph. Such graphs arise frequently in physics, for example in studying crystal structures.)
When we can think of as covering a graph in the -torus . When , covers a graph in the annulus . In either case the cardinality is equal to the number of vertex orbits of , while is the number of edge orbits. The projection map is given by and .
If is a subgroup, then the intermediate covering graph in will be denoted by . The subgroups that we will consider have index , and hence will be a finite -sheeted cover of in the -dimensional torus .
Given a -periodic graph , the Laplacian matrix is defined to be the -matrix , where now is the weighted adjacency matrix with each non-diagonal entry equal to the sum of monomials for each edge between and . (Again, ignoring loops, each diagonal entry of is zero.) The matrix is the same diagonal matrix that we associate to .
The matrix presents a finitely generated -module, the Laplacian module of . The Laplacian (determinant) polynomial is the determinant of . Examples can be found in .
 Let a -periodic graph. Its Laplacian polynomial has the form
where the sum is over all cycle-rooted spanning forests of , and are the monodromies of the two orientations of the cycle.
A cycle-rooted spanning forest (CRSF) of is a subgraph of containing all of such that each connected component has exactly as many vertices as edges and therefore has a unique cycle. The element is the monodromy of the cycle, or equivalently, its homology in . See  for details.
A -periodic graph need not be connected. In fact, it can have countably many connected components. Nevertheless, the number of -orbits of components, henceforth called component orbits, is necessarily finite.
If is a -periodic graph with component orbits , then .
After suitable relabeling, the Laplacian matrix for is a block diagonal matrix with diagonal blocks equal to the Laplacian matrices for . The result follows immediately. ∎
Let a -periodic graph. Its Laplacian polynomial is identically zero if contains a closed component. The converse statement is true when is unweighted.
Conversely, assume is unweighted and no component is closed. Each component of must contain a nontrivial cycle. We can extend this collection of cycles to a cycle rooted spanning forest with no additional cycles. The corresponding summand in 3.1 has positive constant coefficient. Since every summand has nonnegative constant coefficient, is not identically zero.
The Mahler measure of a nonzero polynomial is
(2) When , Jensen’s formula shows that can be described in a simple way. If , , then
where are the roots of .
(3) If , then . Moreover, if and only if is a unit or a unit times a product of 1-variable cyclotomic polynomials, each evaluated at a monomial of (see ).
(cf. ) If is a -periodic graph with nonzero Laplacian polynomial , then
where ranges over all finite-index subgroups of , and denotes the minimum length of a nonzero vector in . When , the limit superior can be replaced by an ordinary limit.
(1) The condition ensures that fundamental region of grows in all directions.
(3) When , the finite-index subgroups are simply , for . In this case, we write instead of .
Before proving Theorem 3.6 we give an example that demonstrates the need for defining graph complexity as we do.
Consider the 1-periodic graph in Figure 2. As before, unlabeled edges are assumed to have weight . Generators for the Laplacian module are indicated. The Laplacian matrix is
The quotient is the finite graph in Example 1. The Laplacian matrix of any is easily described as a block matrix where is replaced by the companion (permutation) matrix for . It is conjugate to a the diagonal block matrix , where is a primitive th root of unity. The matrix is the Laplacian matrix of ,
which has nullity 2. Hence the tree complexity vanishes for every . Nevertheless, by Theorem 3.6 the (torsion) complexity is nontrivial and has exponential growth rate equal to . One can verify directly that the Laplacian subgroup is isomorphic to
We proceed with the proof of Theorem 3.6.
Recall that the Laplacian module is the finitely generated module over the ring with presentation matrix equal to the Laplacian matrix . We let be the additive circle group , and we consider the Pontryagin dual group . We regard as a discrete topological space. Endowed with the compact-open topology, is a compact -dimensional space. Moreover, the module actions of determine commuting homeomorphisms of . Explicitly, for every . Consequently, has a -action .
The pair is an algebraic dynamical system, well defined up to topological conjugacy (that is, up to a homeomorphism of respecting the action). In particular its periodic point structure is well defined.
For any subgroup of , a -periodic point is a member of that is fixed by every element of . The set of -periodic points is a finitely generated abelian group isomorphic to the Pontryagin dual group .
The group is the Laplacian module of the quotient graph . As a finitely generated abelian group, it decomposes as , where is the rank of and denotes the (finite) torsion subgroup. The Pontryagin dual group consists of tori each of dimension . By Theorem 21.1 of , the topological entropy is:
Since the matrix that presents is square, can be computed also as the logarithm of the Mahler measure (see Example 18.7(1) of ). The determinant of is, by definition, the Laplacian polynomial . Hence the proof is complete. ∎
4 Lehmer’s question
In  D.H. Lehmer asked the following yet unresolved question.
Do there exist integral polynomials with Mahler measures arbitrarily close but not equal to 1?
Lehmer discovered the polynomial which has Mahler measure equal to , but he could do no better. Lehmer’s question remains unanswered despite great effort including extensive computer-aided searches [4, 5, 30, 31, 33].
Topological and geometric perspectives of Lehmer’s question have been found . In  we showed that Lehmer’s question is equivalent to a question about Alexander polynomials of fibered hyperbolic knots in the lens spaces . (Lens spaces arose from the need to consider polynomials with .) Here we present another, more elementary equivalence, in terms of graph complexity.
We will say that a Laurent polynomial is palindromic if . By a theorem of C. Smyth  it suffices to restrict our attention to palindromic polynomials when attempting to answer Lehmer’s question.
A polynomial is the Laplacian polynomial of a 1-periodic graph if and only if it has the form , where is a palindromic polynomial.
The Laplacian polynomial of any -periodic graph is palindromic. This is easy to see from the symmetry of the matrix . Since the row-sums of become zero when , it follows also that divides . (Both observations follow also from Proposition 3.1.) Palindromicity requires that the multiplicity of be even. Hence has the form , where is palindromic.
In order to see the converse assertion, consider any polynomial of the form , where is palindromic. A straightforward induction on the degree of shows that we can pair each term with , and then write as a sum of terms . Then is the Laplacian polynomial of a 1-periodic graph, constructed as in the following example. ∎
Multiplying Lehmer’s polynomial by the unit and then by yields which in turn can be written as
This the Laplacian polynomial of a 1-periodic graph . The quotient graph is easily described. It has a single vertex, two edges with weight and two with . The -weighted edges wind twice and six times, respectively, around the annulus in the direction corresponding to . The -weighted edges wind four and five times, respectively, in the opposite direction.
Lehmer’s question is equivalent to the following. Given , does there exist a 1-periodic graph such that
When investigating Lehmer’s question it suffices to consider polynomials of the form , where is palindromic and irreducible. By Proposition 4.2 any such polynomial is realized as the Laplacian polynomial of a 1-periodic graph with a single vertex orbit. As in Example 3.8 the Laplacian matrix of any finite quotient can be obtained from by substituting for the companion matrix for . Hence the nullity of is 1 provided that is not a cyclotomic polynomial (multiplied by a unit), a condition that we can assume without loss of generality. Hence for each (see discussion following Definition 2.1.) Theorem 3.6 completes the proof.
(1) The conclusion of Theorem 4.4 does not hold if we restrict ourselves to unweighted graphs. By Theorem 6.7 below, the Mahler measure of the Laplacian polynomial of any 1-periodic graph with all edge weights equal to 1 is at least 2.
The cyclic 5-fold cover of the graph in Example 4.3 contains the complete graph on 5 vertices, and hence it is nonplanar. Hence we ask:
Is Theorem 4.4 still true if we require that the graphs be planar?
5 Laplacian polynomials of plane 1-periodic graphs
A finite plane graph determines a diagram of a medial link by a simple procedure in which each edge of the graph is replaced by two arcs as in Figure 3. If the graph is unweighted then the resulting diagram is alternating as in Example 5 below. (The reader is invited to sketch the medial link associated to Figure 1. It is a non-alternating boundary link. According to  it was introduced by J. Milnor and is the first link known to have zero Alexander polynomial.)
It is well known that the Laplacian matrix of a finite plane graph is an unreduced Goeritz matrix of the associated link . The reduced matrix, obtained by deleting a row and column, presents the first homology group of the 2-fold cyclic cover of branched over the link. (See [34, 42, 26] for additional details.)
When is a 1- or 2-periodic graph we apply the same construction to produce a diagram of an infinite link . It has a finite quotient diagram modulo the induced - or -action on .
For we regard in an annulus where it describes a link in a solid unknotted torus . The complement is homeomorphic to the , where is the link formed by the union of with a meridian of . The meridian acquires an orientation induced by the infinite cyclic action on . The following result relates the Laplacian polynomial to the Alexander polynomial .
Let be a plane 1-periodic graph and the encircled link . Then
where indicates equality up to multiplication by units in .
We abbreviate by and compute it via Fox calculus. Such a calculation can be done using the link group and augmentation homomorphism that maps a meridian of to and meridians of to . Instead we will make use of , an infinite-index subgroup of . It has a countable Dehn presentation associated to the diagram . The presentation has generators corresponding to bounded regions together with a generator corresponding to one of the two unbounded regions of . (The second unbounded region, called the base region, is labeled zero.) Here is short-hand for the families , indexed by . The generators are arbitrary but fixed orbit representatives. We regard as .
We recall that Dehn generators are related to the more-familiar Wirtinger generators once an orientation of the link is given. A Dehn generator corresponds to a loop that begins at a base-point above the plane, pierces the plane in the region of the Dehn generator, and returns through the base region. While the Dehn presentation requires no orientation of , the restriction of the augmentation homomorphism to is determined only with coefficients modulo 2. For this, we checkerboard color the diagram with black and white in such a way that black regions contain the vertices of . Under the augmentation homomorphism white generators , those corresponding to white regions, map to . Black generators, those belonging to black regions, map to .
Relations of the Dehn presentation correspond to crossings of , as in Figure 4. The relation arising from the crossing at the top of the diagram is while the crossing at the bottom we may write as . Like the generators, the relations come in countable families. Applying Fox calculus to relations, we obtain (top) and (bottom). We denote the collection of Dehn relations by .
Fix a black region of (vertex of ) and identify it with the generator in Figure 4. When we can add the relations involving , the white generators cancel in pairs and we are left with the Laplacian relation of at the vertex. Note that this collection of Laplacian relations is a consequence of .
Consider the dual graph and choose a spanning tree rooted at the base region of . Each edge of crosses a unique edge of , and hence corresponds to a subset of . Using these relations we can express each white generator in terms of black generators. Lemma 5.2 below will show that the remaining relations in are consequences of . Once we use the relations to eliminate white generators and rewrite the remaining relations, we see the presentation of given by the Laplacian matrix . Since the polynomial is the th determinantal invariant of the module divided by , the proof is complete.
Let be a plane 1-periodic graph, and let be the Dehn and Laplacian relations of the link diagram resulting as above from the medial construction. Let be the Dehn relations corresponding to the edges of a spanning tree for the dual graph rooted at the base region of . Then any Dehn relation in is a consequence of .
Any Dehn relation in corresponds to an edge of not contained in . Adding the edge to results in a unique cycle . We proceed by induction on the number of vertices of enclosed by .
If encloses a single vertex, then, as in the proof of Theorem 5.1, the relations corresponding to the edges of add together to give the Laplacian relation at the vertex.
If encloses more than one vertex, then it can be decomposed as a union of cycles and with edges of in common such that each cycle encloses fewer vertices than . The sum of the Dehn relations corresponding to the edges of is equal to the addition of the sums coming from and , the relations corresponding to common edges canceling in pairs. By the induction hypothesis the later is a consequence of .
Question 4.6 might be approached by reversing the process of transforming a plane 1-periodic graph to an encircled link . Given any link with an unknotted component , the Mahler measure of is the Mahler measure of some 1-periodic plane graph. (Neither the extra factor of nor the orientation of will affect the Mahler measure.) However, the classification of Alexander polynomials of links with an unknotted component is a difficult problem that has been only partly solved [21, 11, 41]
6 Unweighted -periodic graphs
We present some results about unweighted -periodic graphs. They will not surprise some experts, but, as far as we know, they have not appeared elsewhere. In particular, we will show that our restatement of Lehmer’s question in Theorem 4.4 is not valid if we restrict ourselves to unweighted graphs.
Graphs considered in this section are unweighted.
We call the limit in Theorem 3.6 the complexity growth rate of , and denote it by . Its relationship to the thermodynamic limit or bulk limit defined for a wide class of unweighted lattice graphs is discussed in .
Denote by a fundamental domain of . Let be the full unweighted subgraph of on vertices . We denote by the corresponding medial link.
If is connected for each , then and have the same exponential growth rates. (See Theorem 7.10 of  for a short, elementary proof. A more general result is Corollary 3.8 of .) The bulk limit is defined by .
The -dimensional grid graph is the unweighted graph with vertex set and edges from to if and , for every . Its Laplacian polynomial is
When , it is a plane graph. The graphs links are indicated in Figure 5 for on left and on right.
The determinant of a link , denoted here by , is the absolute value of its 1-variable Alexander polynomial evaluated at . It follows from the Mayberry-Mott theorem  that if is an alternating link that arises by the medial construction from a finite plane graph, edge weights allowed, then is equal to the tree complexity of the graph (see Appendix A.4 in ). The following corollary is an immediate consequence of Theorem 3.6. It has been proven independently by Champanerkar and Kofman .
Let be a connected -periodic unweighted plane graph, or . Then
(1) In  the authors consider as well more general sequences of links. When , their results imply that:
where is the number of crossings of and is the volume of the regular ideal octohedron.
(2) If Question 4.6 has an affirmative answer then Lehmer’s question becomes a question about link determinants.
Grid graphs are the simplest unweighted -periodic graphs, as the following theorem shows.
Assume that is an unweighted connected -periodic graph. Then .
The conclusion of Theorem 6.4 does not hold without the hypothesis that is connected. Consider the 2-periodic graph consisting of countably many copies of obtained from by removing all vertical edges. Then while .
The following lemma, needed for the proof of Corollary 6.7, is of independent interest.
The sequence of complexity growth rates is nondecreasing.
Doubling each edge of results in a graph with Laplacian polynomial , which has Mahler measure . The following corollary states that this is minimum nonzero complexity growth rate.
(Complexity Growth Rate Gap) Let be any unweighted -periodic graph with Laplacian polynomial . If , then
Although is relatively simple, the task of computing its Mahler measure is not. It is well known and not difficult to see that . We will use a theorem of N. Alon  to show that approaches asymptotically.
(1) , for all .
Asymptotic results about the Mahler measure of certain families of polynomials have been obtained elsewhere. However, the graph theoretic methods that we employ to prove Theorem 6.4 are different from techniques used previously.
Now suppose is a subgraph of consisting of one or more connected components of , such that the orbit of under is all of . Let be the stabilizer of . Then for some , and its action on can be regarded as a cofinite free action of . Consider the limit
where ranges over finite-index subgroups of .
Under the above conditions we have .
Let be any finite-index subgroup of . Then is invariant under . The image of in the quotient graph is isomorphic to .
Note that the quotient of by the action of is isomorphic to , since the orbit of is all of . Since is a -fold cover of and is a -fold cover of , comprises mutually disjoint translates of a graph that is isomorphic to . Hence and
Since as , we have . ∎
Proof of Theorem 6.4. Consider the case in which has a single vertex orbit. Then for some , with , the edge set consists of edges from to for each and . Since is connected, we can assume after relabeling that generate a finite-index subgroup of . Let be the be the -invariant subgraph of with edges from to for each and . Then is the orbit of a subgraph of that is isomorphic to , and so by Lemma 6.9, .
We now consider a connected graph having vertex families , where . Since is connected, there exists an edge joining to some . Contract the edge orbit to obtain a new graph having cofinite free -symmetry and complexity growth rate no greater than that of .
Repeat the procedure with the remaining vertex families so that only remains. The proof in the previous case of a graph with a single vertex orbit now applies. ∎
Proof of Lemma 6.6. Consider the grid graph . Deleting all edges
in parallel to the th coordinate axis yields a subgraph consisting of countably many mutually disjoint translates of . By Lemma 6.9, .
If has an orbit of parallel edges, we see easily that . Otherwise, we proceed as in the proof of Theorem 6.4, contracting edge orbits to reduce the number of vertex orbits without increasing the complexity growth rate. If at any step we obtain an orbit of parallel edges, we are done; otherwise we will obtain a graph with a single vertex orbit and no loops. If is isomorphic to then must be a tree; but then , contrary to our hypothesis. So must have at least two edge orbits. Deleting excess edges, we may suppose has exactly two edge orbits.
The Laplacian polynomial has the form , for some positive integers . Reordering the vertex set of , we can assume without loss of generality that . The following calculation is based on an idea suggested to us by Matilde Lalin.
Using the inequality , for any nonnegative , we have:
Our proof of Theorem 6.8 depends on the following result of Alon.
 If is a finite connected -regular unweighted graph, then
where is a nonnegative function with as .
Proof of Theorem 6.8. (1) The integral representing the logarithm of the Mahler measure of can be written
By symmetry, odd powers of in the summation contribute zero to the integration. Hence
where is a nonnegative function such that . Hence
Theorem 3.6 completes the proof.
One can evaluate numerically and obtain an infinite series representing . However, showing rigorously that the sum of the series approaches zero as goes to infinity appears to be difficult. (See , p. 16 for a heuristic argument.)
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Department of Mathematics and Statistics,
University of South Alabama
Mobile, AL 36688 USA