Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a"large"semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.


Introduction
Let G = (V, E) denote a graph with vertex set V and edge set E ⊆ (V × V )/S 2 , consisting of unordered pairs of elements of V . The edge deck D e (G) of G is the multi-set of isomorphism classes of all edge-deleted subgraphs of G. Harary [13] conjectured in 1964 that graphs on at least four edges are edge-reconstructible, i.e., determined up to isomorphism by their edge deck. This so-called edge reconstruction conjecture is the analogue for edges of the famous (vertex) reconstruction conjecture of Kelly and Ulam that every graph on at least three vertices is determined by its (similarly defined) vertex deck (compare [4]). Many invariants of graphs where shown to be reconstructible from the vertex and/or edge deck, and from the large literature on the subject, we quote the following three sources that are most relevant in the context of our results: (a) vertexreconstruction of the characteristic polynomial (of the vertex adjacency matrix) by Tutte [25]; (b) vertex-reconstruction of the number of (possibly backtracking) walks of given length through a given vertex v ∈ V (which one can specify without knowing the graph G by pointing to the element G − v of the vertex deck) by Godsil and McKay [11]; (c) edge reconstruction for graphs with average degree d ≥ 2 log 2 |V | by Vladimír Müller [22], improving upon a method of Lovász [19].
Following the discussion by McDonald in [20], the edge reconstruction conjecture should also hold for multi-graphs. Since disconnected (multi-)graphs are reconstructible ([4], 6.14(b), [20], If r ∈ Z ≥1 , the entry (T r ) e 1 , e 2 is the number of non-backtracking walks of length r on G that start in the direction of e 1 and end in the direction of e 2 . As for the usual adjacency matrix, graphs can have the same eigenvalues for T without being isomorphic ( [24], Chapter 21).
The matrix T is related to the Ihara zeta function ζ G of G [16], defined as the following analogue of the Selberg zeta function from differential geometry (cf. [24], Part I): where the product runs over classes of non-backtracking tailless closed oriented prime walks p in G (where "class" refers to not having a distinguished starting point, and "prime" refers to not being a multiple of another walk), and ℓ(p) is the length of p. The function ζ G (u) is a formal power series in u, but it is also convergent as a function of the complex variable u for |u| sufficiently small. We have an identity ( [2], II.3.3) showing that ζ G has an analytic continuation to the entire complex plane as a rational function with finitely many poles. If one so wishes, one may take Equation (2) as a definition of ζ G ; in this paper, the original definition as in Equation (1) will not play any rôle. We will prove the following: Theorem A. Let G denote a graph of average degree d. The following are edge-reconstructible: (i) If d ≥ 4, the Ihara zeta function ζ G of G, i.e., the spectrum of the edge-adjacency matrix T ; in particular, the Perron-Frobenius eigenvalue λ PF of T ; (ii) If d ≥ 4, the number N r of non-backtracking closed walks on G of given length r; (iii) If d > 4, the functions (a) N r : D e (G) → Z that associates to an element G−e of the edge deck of G the number N r (e) of non-backtracking closed walks on G of given length r passing through e; (b) M r : D e (G) → Z that associates to an element G−e of the edge deck of G the number M r (e) of non-backtracking (not necessarily closed) walks on G of given length r starting at e (in any direction); (iv) If d > 4, the function D e (G) → R 2 (where R 2 is the set of unordered pairs of real numbers) that associates to an element G−e of the edge deck the unordered pair {p e , p e } of entries of the normalized Perron-Frobenius eigenvector p of T ; (v) If d > 4, the function F r : D e (G) → Z that associates to an element G − e of the edge deck D e (G) of G the number of non-backtracking closed walks on G of given length r that pass through e in both directions at least once.
The statements (iii)-(v) in the theorem make sense, since if G − e ∼ = G − e ′ , the functions turn out to have the same value at e and e ′ (cf. Remark 5.2).
We indicate briefly how to prove these results. Deleting an edge from the graph corresponds to deleting two rows and columns from the matrix T (namely, corresponding to the two possible orientations on the edge). The proof of (i) starts with a lemma on the combinatorial reconstruction of the top half of the coefficients of det(λ − T ) from second derivatives of 2 × 2-minors of T (Section 1). The next step in the proof is to exploit certain relations between the coefficients in det(λ − T ) which arise from a formula of Bass that relates det(λ − T ) to a polynomial of degree 2|V |-there are enough relations to reconstruct all coefficients if the stated condition on the average degree holds (Section 2; in a sense, this is an analogue of the "functional equation" for the Ihara zeta function of a regular graph). Part (ii) follows by expressing the formal logarithm of the Ihara zeta function as a counting function for such closed walks. Alternatively, one may take this expression as a starting point of the proof, reduce the problem in (i) to that of counting closed walks of length < |E|, and use Kelly's Lemma. The proof of (iii) uses the Jordan normal form decomposition for the matrix T , the non-vanishing of an associated "confluent alternant" determinant and the fact that T has a "large" semi-simple part to reduce the counting problem to length < |E|, which then again is done by purely combinatorial means. In case of non-closed walks, this also involves identities based on decomposition of walks into closed and non-returning walks. On the way, we prove some further spectral properties of T , e.g., that it is symmetric on a Π |E| -Pontrjagin space (Proposition 3.2), and an explicit description of its ±1 eigenspaces in terms of certain spaces of cycles on the graph (Propositions 4.2 and 4.5). We also point out that the presence of end-vertices in the graph leads to a non-semi-simple T -operator (Proposition 3.3). Part (iv) follows from known behavior of the Cesàro averages of powers of non-negative matrices. Finally, part (v) follows by using an identity of Jacobi for 2 × 2-sub-determinants.
Two open problems that arise from the proofs and that we want to highlight are the following: (a) can the Ihara zeta function ζ G be reconstructed from the (multi-)set {ζ G−e : e ∈ E} of Ihara zeta functions of edge-deleted graphs?; (b) for |E| ≥ 2, is T semi-simple if and only if G has an end-vertex?.
We finish this introduction by listing some applications.
As we explain in [7] (cf. also [9]), the invariants that we have reconstructed play a central role in the measure-theoretical study of the action of the fundamental group on the boundary of the universal covering tree of the graph. More precisely, the fundamental group Γ of G, a free group of rank the first Betti number b > 1 of G, acts on the boundary of the universal covering tree of G. This dynamical system "remembers" only b, since it is topologically conjugate to the action of the free group of rank b on the boundary of its Cayley graph. However, the graph is uniquely determined by a measure on the boundary, namely, the pull-back of the Patterson-Sullivan measure for the action of Γ on the boundary. For this measure, the boundary has Hausdorff dimension log λ, where λ is the Perron-Frobenius eigenvalue of T , and the measure itself is expressed on a set of generators for Γ in terms of λ, the entries of the Perron-Frobenius eigenvector of T , and the lengths of the loops corresponding to the generators In [18], the operator T is used for spectral algorithms that find clustering in large graphs. This is a hard problem if the graphs under consideration are sparse with widely varying degrees, and the authors argues that use of the operator T outperforms classical algorithms based the spectrum of the adjacency or Laplacian operator. Since the input for their clustering algorithm consists of the two leading eigenvalues of T , our main theorem shows reconstruction of this input (if d ≥ 4).
In the theory of evolution of species, it has recently been argued that evolutionary relations are not always tree-like [1]. Thus, the phylogenetic reconstruction problem should be considered in the context of general multigraphs, rather than the more traditional case of trees, and our theorem gives a theoretical underpinning for this more general question of reconstruction.

A lemma on polynomial coefficients
Proof. Let m = |E|, and order the rows and columns of the 2m × 2m matrix T G so that for all e ∈ E, the two orientations e and e label adjacent columns and rows. Set and consider the multi-variable polynomial By construction, P G has at most degree 2 in each of the individual variables, and after specialisation of all variables to the same λ, we find det(λ − T G ). The theorem follows by applying the following lemma to P = P G , observing that the formula for the expansion of a determinant by (2m − 2) × (2m − 2)-minors implies which we use iteratively to replace in (5).
Proof. Since the statement is linear in P , it suffices to prove (5) if P is a monic monomial and d = deg P (λ, . . . , λ) (since for other d, the left and right hand side are both zero), when the left hand side is 1. Suppose that such a monomial P contains exactly k quadratic factors λ 2 i . Since we assume d > m, we have k ≥ 1, and since P has degree d, we also have k ≤ d/2. Then the right hand side equals

A formula of Bass and reconstruction of ζ G
If the graph G under consideration is (q +1)-regular for some q ∈ Z ≥2 (when the reconstruction problem is easy), the Ihara zeta function satisfies functional equations, for example ([2], II.3.10) This implies "palindromic" relations between the top m and bottom m coefficients of ζ −1 G (u), so that reconstruction of half the coefficients would be enough. In the general (irregular) case that we consider here, there is no such functional equation, but as a substitute for finding relations between the coefficients, at the cost of assuming a certain minimal average degree, we will use an identity of Bass ([2], II.1.5), stating that where A is the adjacency matrix of G and Proof. Set P (λ) = det(λ − T ), and A(λ) = (λ 2 − 1) |E|−|V | . The identity of Bass becomes P (λ) = A(λ)B(λ). All coefficients [λ i ]A are easily computable and depend only on |E| and |V |; also note that for even i, they are non-zero. Now |V | and |E| are edge-reconstructible as |V | = |V − e| and |E| = |E − e| + 1 for any e ∈ E. The previous theorem implies that the coefficients [λ k ]P are edge reconstructible for k = |E| + 1, . . . , 2|E|. We will use this to reconstruct the coefficients [λ d ]B for d = 2|V | − |E| + 1, . . . , 2|V |. We use the formula recursively. For k = 2|E| we find the relation . . and note that in each step corresponding to [λ 2|E|−j ]P we find recursively that the only unknown term in the above sum is ]P is the highest coefficient which is not reconstructed by the previous theorem.
Proof. We first observe that In Section 5, we will give another proof of Theorem A(i) that avoids Lemma 1.3, but has the disadvantage of not leading directly to the formula from Theorem 1.2 for the coefficients in terms of coefficients corresponding to edge-deleted subgraphs.
In analogy to the question whether the characteristic polynomial of G is determined uniquely by those of its vertex deleted subgraphs [12], one may ask Proof. The number |e| of distinct edges in e, is determined by the degree of ζ −1 G−e . The formula in Theorem 1.2 can be rewritten as for d > |E|, which is reconstructible from Z (G). As in Lemma 2.1, we can then also reconstruct all coefficients of ζ −1 G , as soon as 2|V | − |E| + 1 < 1, i.e., d > 4.
If Z (G) uniquely determines det(D−1), then one may replace the bound d > 4 in this theorem by d ≥ 4.

Remark 2.7.
We list some properties that have been shown to be determined by ζ G (hence for d ≥ 4 are edge-reconstructible by our main theorem; but the edge-reconstructibility of these invariants was already known from Kelly's Lemma): (

Symmetry of the Bass-Hashimoto edge adjacency operator
The matrix T is not symmetric in general: for a graph G in our sense, this only happens if G is a "banana graph" consisting of two vertices connected by several edges (since T being a symmetric matrix means that T e 2 , e 1 = 1 whenever T e 1 , e 2 = 1). However, T does have a certain symmetry. where J is a block matrix The signature of this form is (|E|, |E|), and (R 2|E| , , ) is a finite dimensional Kreȋn space (i.e., an indefinite metric space, compare [3]).
An eigenvalue is called semi-simple if its algebraic and geometric multiplicity are equal.
Proposition 3.2. The operator T : R 2|E| → R 2|E| is symmetric for an (indefinite) metric ·, · of signature (|E|, |E|). Its generalized eigenspaces are mutually orthogonal for this metric, and T has at most |E| non-semi-simple eigenvalues.
Proof. Observe that T e 1 , e 2 = T e 2 , e 1 for all e 1 , e 2 ∈ E. By enumerating the rows and columns of T as e 1 , . . . , e |E| , e 1 , . . . , e |E| , we see that T is of the form Being of this form is equivalent to the fact that T satisfies an equation Equation (7) Not every , -symmetric matrix in a Kreȋn space is diagonalisable (e.g., the matrix I |E| I |E| 0 I |E| is J-symmetric but not semi-simple). It is easy to construct examples of graphs for which T is not semi-simple, if we temporarily drop our assumption that the graph has no end-vertices: Proof. If e is an oriented edge that ends in an end-vertex (so T e , * = 0 for all * ∈ E), then e∈ ker T , and if e 1 is an oriented edge with t( e 1 ) = o( e) (which exists by connectedness and since |E| > 1), then e 1 ∈ ker T 2 − ker T . Question 3.4. Give necessary and/or sufficient criteria for a (multi-)graph G to have a semi-simple edge-adjacency operator T . More specifically, is the presence of end-vertices the only obstruction to semi-simplicity?

The ±1-eigenspaces of the Bass-Hashimoto edge adjacency operator
In the next two propositions, we show that T has a "large" semi-simple quotient described in terms of the cycle space of G. Since we will use concepts and notation from the (short) proof, we outline it here: Proof. Since b 1 > 1, the multiplicity of the eigenvalue 1 in the characteristic polynomial of T is equal to the first Betti number b 1 ([2], II.5.10(b)(i) [14], 5.26). It follows that ker(1 − T ) has dimension ≤ b 1 . Therefore, it suffices to prove that the map ϕ is well-defined and injective.
To show well-definedness of the linear map ϕ, fix an induced cycle c = e 1 + · · · + e r . Assume that we read the indices of the edges e i occuring in c as indexed by integers modulo r. for some a * ∈ C, then a e e = 0, so only the zero cycle is mapped to zero.
Next we consider the eigenspace of eigenvalue −1.

Notation 4.4.
The integer p is defined by p = 0 if G is bipartite and p = 1 otherwise. Let H + 1 (G, C) denote the subspace of H 1 (G, C) generated by cycles of even length.
We have H 1 (G, C) = H + 1 (G, C) ⊕ C p . Indeed, a graph is bipartite if and only if all cycles are even ([10], 1.6.1), and if the graph is not bipartite, let c 1 , . . . , c r , c r+1 , . . . c b 1 denote a basis for its cycle space based at a common vertex v 0 , in which the first r cycles are even and the remaining are odd. Then is a basis in which the first b 1 − 1 cycles are even and the final one is not. . It suffices to prove that the map ψ is well-defined and injective. For welldefinedness, fix an induced even cycle c = e 1 + · · · + e r as before. Without loss of generality, we can assume κ c (e j ) = (−1) j . Then, using the notation for "bushes" from the proof of Proposition 4.2, we find so ψ is well-defined. The injectivity of ψ follows again from the linear independence of the elements e, e (for e ∈ E). There are examples (such as the complete 4-graph with one edge deleted [24], Example 2.8) in which all other eigenvalues of T , apart from ±1, are simple (and semi-simple). This shows that one cannot expect a more general statement than 4.6 concerning multiplicities of eigenvalues of T .

Reconstruction of closed non-backtracking walks
For a positive integer r, the entry of T r at place e 1 , e 2 is the number of non-backtracking walks that start in the direction of the oriented edge e 1 and end at the oriented edge e 2 . Let N r ( e) = T r e , e denote the number of closed such walks through an oriented edge e∈ E. Observe that by symmetry ("walking backwards"), N r ( e) = N r ( e). For an unoriented edge e ∈ E, N r (e) = 2N r ( e) (for any choice e of orientation on e), denotes the number of (oriented) non-backtracking closed walks that pass through e. The total number of non-backtracking (unoriented) closed walks of length r in G is where tr denotes the trace of a matrix.

Theorem 5.1 (Theorem A(ii)). Let G denote a graph of average degree d ≥ 4; then the number of non-backtracking closed walks on G of given length is edge-reconstructible.
Proof. The claim follows directly from the formal power series identity and part (i) of the theorem.
We now refine this result, in analogy with the vertex situation studied by Godsil and McKay in [11] (but our proofs are rather different, since we do not have a semi-simple operator and we cannot rely on reconstruction results for complementary graphs).

Remark 5.2.
We define the value of N r (and other similar functions) at an element H = G − e ∈ D e (G) of the edge deck to be equal to N r (e). Since D e (G) is a multiset, it is possible that G − e ∼ = G − e ′ for two different edges e and e ′ . Our methods of proof imply that the value N r (e) only depends on the isomorphism type of H, not on the edge e, and thus, N r is well-defined on the edge deck. Proof. As a first step, we use the Jordan normal form of T to prove the following: Proof of Lemma 5.4. Suppose that T has N distinct eigenvalues λ 1 , . . . , λ N . Let m i denote the multiplicity of λ i . Suppose that λ i occurs in ℓ i different Jordan blocks, and let µ i,j denote the size of the j-th such block (j = 1, . . . , ℓ i ), so that m i = j µ i,j . Let P denote the matrix whose columns are a complete set of generalized eigenvectors for T , then T = P ΛP −1 , where Λ is a Jordan normal form of T . Fix an (oriented) edge e. All vectors will depend on e, but, for readability, we will mostly suppress it from the notation. If x e is the 2|E|-column vector with a 1 in place e and 0 elsewhere, then where v = x ⊺ e P and v ′ = P −1 x e . Working out the powers of the Jordan normal form, we find that for some constants Set new variables w i,j,k = 0 when k ≥ µ i,j ; with this convention, we can replace the third summation in (10) by k = 0, . . . , M i − 1, independent of j. Hence we can collect terms in j, to find that there exists constants y i,k such that namely, The set of equations (11) can be written in matrix form as where Y is a column vector consisting of y i,k , N is a column vector with entries N i ( e) for i = 0, . . . , M − 1, and V is the M × M -matrix given as concatenation Note that V is edge-reconstructible by our reconstruction of the spectrum of T . If T is semisimple, this is a classical Vandermonde matrix. In general, it is a Vandermonde matrix with inserted columns corresponding to powers of the nilpotent part of T ; it is the matrix consisting of generalized eigenvectors for the companion matrix of the characteristic polynomial of T and historically known as a "confluent alternant" [17]. We have (loc. cit., Formula (14)) and hence V is invertible. Therefore, Y is uniquely determined by N, and N r (e) is uniquely determined for all r by its values for r ≤ M − 1.
As a second step, we prove that for d > 4, M − 1 < |E|. Indeed, recall from Corollary 4.6 that T has semi-simple eigenvalue λ 1 = +1 with multiplicity |E| − |V | + 1 and semi-simple eigenvalue λ 2 = −1 with multiplicity at least |E| − |V |. Hence M 1 = M 2 = 1 and the number M satisfies Since we assume d = 2|E|/|V | > 4, we have M − 1 < |E|. Finally, we show how to reconstruct N r (e) for r < |E|. Suppose that G i is the set of isomorphism classes of graphs with i edges. Given a graph H, let P r (H) denote the number of distinct closed non-backtracking walks of length r on H that go through every edge of H (possibly multiple times, with no preferred starting edge). Let S(H, G) denote the number of subgraphs of G isomorphic to H. For r < |E|, we have , since H has less than |E| edges, the right hand side is reconstructible, hence so is the left hand side.
This finishes the proof of the theorem that N r ( e) is edge-reconstructible for all r.
Proposition 5.5. If G is bipartite of average degree d ≥ 4, the function N r is edge-reconstructible for all r > 0.
We now give another proof of part (i) of Theorem A along the lines of the previous proof, which has a more combinatorial flavour and avoids using Lemma 1.3 (but does not lead directly to the inductive formula from Theorem 1.2). for some polynomial D + (u) of degree 2|V | − 1 with D + (0) = 0. Plugging this into the generating series (8) and take logs, we find

Second proof of Theorem
It follows that we know the entire polynomial det(1 − T u) as soon as we know D + (u), which happens as soon as we know N r for all r ≤ 2|V | − 1. With d = 2|E|/|V | ≥ 4, we need to reconstruct N r for r < |E|. But this can be done using Kelly's Lemma, as follows: where G i , P r and S(H, G) are as in the above proof of Theorem A(iii).
For e ∈ E, let F r (e) denote the number of closed non-backtracking walks that pass through e in both directions at least once. Then F r (e) = 2F r ( e), where for an oriented edge e∈ E, F r ( e) is the number of closed non-backtracking walks that start at e and pass through e at least once.
The edge adjacency matrix T G−e of G − e is the matrix T in which the rows and column corresponding to the edges e and e have been removed. Let T [e 1 , e 2 ] denote the 2 × 2 matrix in which only the elements in column/row e 1 and e 2 are preserved. In this situation, Jacobi's identity (generalising from 1 × 1 minors to 2 × 2 minors the more familiar formula for an inverse matrix in terms of determinant and adjugate; see e.g., [5]) states that The left hand side of this equation is reconstructible by part (i). Since we find that the right hand side equals using the expression for F r ( e) from (13). Since N r ( e) is edge-reconstructible, we conclude that the function F r ( e), and hence F r (e), is edge-reconstructible.
Similar to Proposition 5.5, we get Proposition 5.7. If G is bipartite of average degree d ≥ 4, the function F r is edge-reconstructible for all r > 0.

Reconstruction of non-closed non-backtracking walks
We now consider the case of not-necessarily closed non-backtracking walks: Proof. Let M r ( e) denote the number of non-backtracking walks of length r that start in the direction of e (but do not necessarily return to e). Then, similarly to the expression derived for N r ( e) in the previous proof, we find where 1 is the 2|E|-column vector consisting of all 1's and y ′ i,k is an expression similar to y i,k in the previous proof, but with the role of v ′ taken by 1. Now (where we have indicated the dependence of y ′ i,k on the oriented edge e in the subscript) is the number of non-backtracking walks of length r that start at e in any direction. The same reasoning as in the previous proof shows that is suffices to reconstruct M r (e) for r < |E|; namely, we find a matrix equation where Y ′ is a column vector consisting of y ′ e ,i,k + y ′ e ,i,k , M is a column vector with entries M i (e) for i = 0, . . . , M − 1, and V is the same (invertible) matrix as in the previous proof. This shows that Y ′ , and hence M r (e) for all r, is determined by M r (e) for r ≤ M − 1 < |E|.
Let W r (e) denote the total number of walks through the edge e. This number is reconstructible by Kelly's Lemma for r < |E|, since where Q r (H) is the number of (not necessarily closed) walks of length r that pass through every edge of H.
Let O r ( e) denote the number of walks of length r starting at e that never return to e (but might go though e), and let O r (e) = O r ( e) + O r ( e) denote the number of walks starting in e but never return to e in the same direction. We call these non-returning walks. We then have the following relations (similar to the ones for vertex walks discussed in [11], Formula (1)): (1) Every walk of length r through e decomposes as a non-returning walk of length i into e, then a closed walk of length j through e, followed by a non-returning walk of length k starting at e, for r + 2 = i + j + k (see Figure 2). Hence If we write these relations (14) and (15)  Since we have already reconstructed N j (e) for all j and W i (e) for all i < |E|, we can use this formula to reconstruct recursively the values M r (e) for all r < |E|. This suffices to reconstruct M r (e) for all integers r.
Similar to Proposition 5.5, we get Proposition 6.2. If G is bipartite of average degree d ≥ 4, the function M r is edge-reconstructible for all r > 0. as in Equation (16) below. The result follows from Perron-Frobenius theory for non-negative matrices (see, e.g., section 8.3 in [21]), as follows. Formula (7) implies an equivalence between left and right eigenvectors for T , as follows:

Reconstruction of the Perron-Frobenius eigenvector of T
T Therefore, the Cesàro averages of T give x ⊺ e T r x e λ r PF = p e p e = π e .
Similarly, we have Hence the numbersσ e and π e can be reconstructed from D e (G), since the left hand side of the above formulas (17) and (18) can. Since the entries of the Perron-Frobenius eigenvector are all non-negative, we find that α is positive. Adding up all terms in (18), we find that e∈Eσ e = α 2 , hence α ≥ 0 is determined, and so also σ e =σ e /α is edge-reconstructible. The final statement follows since the elements of the unordered pair {p e , p e } are the roots of x 2 − σ e x + π e = 0.
Similar to Proposition 5.5, we get