An Ambarzumian type theorem on graphs with odd cycles

We consider an inverse problem for Schr\"odinger operators on a connected equilateral graph $G$ with standard matching conditions. The graph $G$ consists of at least two odd cycles glued together at a common vertex. We prove an Ambarzumian type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential has to be zero.


Introduction
The aim of this paper is to make a statement in [25] more accurate and general.The addressed problem originates from a work of Ambarzumian [2] on reconstruction of a differential operator from its eigenvalues.The material from the theory of Sturm-Liouville equations is summarized in [5,14,8,15], previous results for graphs are [22,6,19,10].Another source of the problem is the so-called quantum graphs, i.e., differential operators on graphs [3,23,18]).A third component is the calculation of spectral determinants (or alternatively functional determinants or characteristic functions) [9,16,7,21,1,11,12,13,24,20].For a more detailed discussion of these results see the Introduction in [17].

Results and discussion
Let r ≥ 2 and consider r cycle graphs C 1 , C 2 , . . ., C r with odd cycle lengths n 1 , n 2 , . . ., n r (n j = 1 is also possible).Let the vertices of C j be v j0 , . . ., v jn j = v j0 , and let us form the graph G as the union of C j 's, identifying the vertex v j0 for all j.We shall say that G is a graph consisting of r ≥ 2 odd cycles glued together at a common vertex.The edge of G between v j k−1 and v jk is sometimes denoted by e jk ; however, when the particular location of the edges are not important, we shall refer to them as e 1 , e 2 , . . ., e |E| .
Choosing an arbitrary orientation, we parametrize each edge with x ∈ [0, 1], and consider a Schrödinger operator with potential q j (x) ∈ L 1 (0, 1) on the edge e j and with Neumann (or Kirchhoff) boundary conditions (sometimes called standard matching conditions), i.e., solutions are required to be continuous at the vertices and, in the local coordinate pointing outward, the sum of derivatives is zero.More formally, consider the eigenvalue problem −y ′′ + q j (x)y = λy (2.1) on e j for all j with the conditions if e j and e k are incident edges attached to a vertex v where κ = 0 for outgoing edges, κ = 1 for incoming edges (and can be both 0 or 1 for loops); and in every vertex v (loops are counted on both sides).
If the lengths of the odd cycles are all 1, i.e., the cycles are all loops, then the statement reduces to that of Theorem 2.1 in [25], which states the following: Suppose G is a flower-like graph, i.e., a single vertex attached r loops of length 1.For k = 1, 2, . .., let m k be a sequence of integers with lim m k = +∞.If eigenvalues are nonnegative, λ k = (2m k + 1) 2 π 2 are eigenvalues with multiplicities (r − 1), where m k is a strictly ascending infinite sequence of positive integers, then q j (x) = 0 a.e. on [0, 1], for each j = 1, 2, . . ., r.We have to require r ≥ 2 for the consequence to hold.

Calculation of the spectral determinant
Denote by c j (x, λ) the solution of (2.1) which satisfies the conditions c j (0, λ) − 1 = c ′ j (0, λ) = 0 and by s j (x, λ) the solution of (2.1) which satisfies the conditions s j (0, λ) = s ′ j (0, λ) − 1 = 0.Each y j (x, λ) may be written as a linear combination Then y j (0, λ) = A j (λ) is the same on each outgoing edge; hence as in [17], we index the functions A(λ) by vertices, and then 25,17]).The coefficients A v and B j form a (|V | + |E|)-dimensional vector, which satisfies |V | Kirchhoff conditions at the vertices and |E| continuity conditions at the incoming ends of edges, namely, for all v ∈ V (G), e j :...→v where in the first sum v j denotes the starting point of e j ; and for all e j ∈ E(G), if e j points from u to v (see eq. ( 2.3) and (2.4) in [17]).
The matrix M of this homogeneous linear system of equations has a special structure, the description of which we repeat from [17].Namely, , where • A is like an adjacency matrix; , the sum is taken on edges pointing from u to v; • B and C are like incidence matrices; The determinant of the matrix M is the so-called spectral determinant of the problem (2.1)-(2.3).
Example 1.Consider a flower-like graph, i.e., a single vertex with r loops.Then corresponding to formula (2.9) in [25].
The elements of M have the following asymptotics for λ = (2k + 1) 2 π 2 + d + o(1) (see [6] eq. ( 2.3) or [19] Lemma 3.1): Remark.Using these asymptotics, we get This is a special case of (4.1) and of (4.2) below.We shall use the following terminology: a graph is a saturated forest if every component has exactly one cycle.The saturated forest is odd, if it does not contain even cycles.Note that in a saturated forest, the number of edges and vertices are equal.

The proof
Lemma 4.2.The determinant of the (unoriented) incidence matrix of an odd saturated forest is ±2 κ where κ denotes the number of components.
Proof.The incidence matrix (see in the Glossary) is a direct sum of that of the components, hence it is enough to prove the statement for connected graphs.It is true for odd cycles as well as for a single vertex with a loop.If the graph is not a single cycle or a single loop, there is at least one vertex with only one incident edge.Leaving this vertex out from the graph does not change the absolute value of the determinant of the incidence matrix.This can be repeated until we reach a single cycle or a single loop.
Proof of Theorem 2.1.
Let us introduce Q j = n j l=1 1 0 q jl .For a fixed m substituting d = Qm nm to (4.4), we get 1 Let us denote the operator of the eigenvalue problem (2.1)-(2.3)by L. ϕ,Lϕ ϕ,ϕ ≥ λ 0 = 0 and equality holds if and only if ϕ is an eigenfunction of L. It follows that the constant 1 must be an eigenfunction corresponding to the eigenvalue 0. Substituting this to (2.1) gives q(x) = 0.

Glossary
A walk W in a graph is an alternating sequence of vertices and edges, say X 0 , e 1 , . . ., e l , X l where e i = X i−1 X i , (0 < i < l).The length of W is l.This walk W is called a trail if all its edges are distinct.A path is a walk with distinct vertices.A trail whose end vertices coincide (a closed trail) is called a circuit.To be precise, a circuit is a closed trail without distinguished endvertices and direction, so that, for example, two triangles sharing a single vertex give rise to precisely two circuits with six edges.If a walk W = X 0 , e 1 , . . ., e l , X l is such that l > 3, X 0 = X l , and the vertices X i , 0 < i < l, are distinct from each other and X 0 , then W is said to be a cycle ( [4], p.5).
The incidence matrix of a graph has a row for each vertex and a column for each edge, and is defined as if e j is not incident to v i , 1 if e j is not a loop and incident to v i , 2 if e j is a loop at v i . (5.1)

Lemma 4 . 3 . 1 √λ 1 √λ
If λ = (2k + 1) 2 π 2 + O(1), then the determinant of a |V | × |V | submatrix of C (and of B) is ±2 κ + o( 1 √ λ ) if the rows in C (columns in B) corresponds to the edges of an odd saturated forest, and o( ) otherwise.Proof.Leaving out the o( 1 √ λ ) terms from the submatrix we make only o( ) error in its determinant.What we get is the negative of an incidence matrix of a subgraph with |V | vertices and |V | edges.If the subgraph contains an even cycle, then the corresponding rows are dependent.If a component of the subgraph contains more than one odd cycles or loops (and no even cycles), then consider a path between the two; the corresponding rows are also dependent.The number of cycles is equal to the number of components hence if the determinant is not o( 1 √ λ ), the rows in C must correspond to the edges of an odd saturated forest.The proof for B is similar.

Theorem 4 . 4 .Lemma 4 . 5 .
If λ = (2k + 1) 2 π 2 + O(1), then det M = (−1) |V | τ 4 κ(τ ) e j / ∈τ √ λs j (1, λ) + o(λ − |E|−|V |+1where the sum is taken for all odd saturated forest τ of G. Proof.The main terms in the Laplace expansion are those which contain exactly (|E| − |V |) elements from D. The product of a fixed set of (|E| − |V |) elements in D is weighted by the determinant of the respective minor, with all other elements of D substituted by zero.The remaining rows in C and columns in B look like an unordered incidence matrix of the graph τ spanned by the remaining |V | edges.Then the determinant of the minor is (−1) |V | times the square of the determinant of the incidence matrix of τ .The total multiplicities of the eigenvalues λ = (2k + 1) 2 π 2 + O(1) are exactly |E| − |V |.