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Eigenvalue asymptotics for the damped wave equation on metric graphs

We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high-frequency abscissas, whose location is solely determined by the averages of the damping terms on each edge. We further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate.


Introduction
The simplest model for an inhomogeneous damped vibrating string is given by the equation complemented with an initial and boundary conditions at the end points. Although quite simple, the explicit dependence of the damping on the space variable makes the problem genuinely non-selfadjoint and is responsible for several interesting properties which have received much attention in the literature within the last twenty years -see [BF09,GH11] and the references therein. One particular aspect which is of interest is the characterisation of the spectrum and, within that scope, the behaviour of the high frequencies.
In the case of equation (1) with vanishing b, it was first suggested by the formal calculations in [CFNS] that the real part of these high frequencies would cluster around minus the average of the damping term as the corresponding imaginary parts converge to infinity. This was proven rigorously in [CZ94] The where the first two terms in the asymptotic expansion of the eigenvalues were computed, while the complete asymptotic expansion was obtained in [BF09] for the first time.
In the case of more complex structures involving several segments which are joined at the endpoints, it makes sense to model each component by an equation of the form (1) with corresponding potentials and damping functions and have either a coupling at the common vertices or some boundary condition such as Dirichlet at the isolated endpoints. Indeed, there is a vast literature on the topic of waves on networks of strings as may be seen in the recent review [Z12]. However, most of this work revolves around the observability and controlability of such problems and, as far as we are aware, the damped case has not been considered in the literature. In particular, this means that in many cases the problem may still be reduced to the study of a self-adjoint (vector) operator.
A different starting point which is also connected to the undamped case comes from the quantum graph literature where the resulting problem, although including the explicit dependence on the spatial variable via a potential playing a similar role to b in equation (1) above, is still reduceable to a self-adjoint operator for a wide variety of coupling conditions (see e.g. [Kuc04,KS99,RS12]).
The main object of study in the present paper is the wave equation on graphs with potential and damping functions depending explicitly on the space variable and with coupling conditions at the vertices. We are interested in understanding the asymptotic behaviour of the associated spectrum and, in particular, the counterpart for the case of graphs of the result on the spectral abscissa mentioned above for one segment. This is related to the rate of decay of high frequency modes on such structures and is therefore an important issue in applications. We remark that, as already noted in [Z12], this behaviour depends in a subtle way on topological and number theoretic properties of the network, without it being possible to understand the global dynamics by simply studying each component in isolation from the rest.
Our main contribution here is to describe the possible asymptotic distributions of the high frequencies and show that, under certain conditions on the edge lengths and the coupling conditions, there exists at most a finite number of values around which the real parts of the eigenvalues may cluster, as the imaginary parts grow to infinity -see Theorems 3.4 and 4.3, the results in Section 6 and the examples in the last section. In particular, we show that under such conditions these high frequency spectral abscissas are the same as those obtained for the same graph where the damping terms are replaced by their averages on each edge.
While in the case of N equilateral edges we show that the number of such abscissas is at most 2N and that, if the graph is bipartite (does not have loops of odd length), this number cannot be larger than N , we also show that these results are sharp in the sense that for certain classes of graphs there exist damping terms yielding these maximal number of abscissas. If we drop the assumption that all edges have equal lengths and assume instead that they are comensurate, we give examples which indicate that the number of these abscissas may now be as large as desired.
The damped wave equation on metric graphs may thus be viewed as a bridge in complexity between the scalar damped wave equation in one and two spatial dimensions. While in the former case there is only one high frequency spectral abscissa corresponding to the averaging of the damping on a segment, in the two dimensional case there are several different behaviours for the high frequencies corresponding to the different trajectories on the domain [BLR92, AL03, Sjö00]. On a metric graph, these trajectories are restricted to the edges, with the damping being averaged along each edge and these averages then being combined via the topology of the graph.
The paper is structured as follows. In the next section we describe the model and the corresponding notation. The third section presents some basic asymptotic properties of the secular equation and vertex scattering matrix for eigenvalues with high imaginary part. The number and location of high frequency abscissas are studied in Sections 6 and 4. The method of pseudo orbit expansion (section 5) is used for proving parts of Theorem 6.6 which in particular states that for obtaining the location and number of spectral abscissas we may replace the damping and potential terms on each edge by their averages. In Section 7 several examples are provided illustrating the previous results.

Description of the model
We consider a compact metric graph Γ with N < ∞ finite edges {e j } N j=1 of lengths {l j } N j=1 . On each edge e j a linear damped wave equation of the form is considered, with both the damping functions a j (x) and the potentials b j (x) real and bounded. The above set of partial differential equations can be rewritten in a more elegant way where I is the N × N identity matrix, A and B are diagonal N × N matrices with entries a j and b j , respectively, w 0 is the vector of functions w j and w 1 its time derivative. The Ansatz w j (t, x) = e λt u j (x) allows us to rewrite equation (2) as and therefore reformulate the time dependent problem as a spectral problem. More precisely, in this way we are led to the operator with domain consisting of functions (ψ 1 (x), ψ 2 (x)) T with components of both ψ 1 and ψ 2 in W 2,2 (e j ) for the corresponding edge and satisfying suitable coupling conditions at the vertices. Now we clarify the term "suitable coupling conditions"; we construct all operators H satisfying the above properties, for which iH A=0 is selfadjoint. We define ψ = (ψ 1 , ψ 2 ) T , φ = (φ 1 , φ 2 ) T , denote by (·, ·) the L 2 (Γ) scalar product linear in the second argument and antilinear in the first one and by ((·, ·)) the L 2 (Γ) ⊕ L 2 (Γ) scalar product. For the condition for selfadjointness of iH we obtain and hence 0 = (ψ 1 , φ 2 ) + (ψ 2 , φ 1 ) + (ψ 2 , Bφ 1 ) + (ψ 2 , φ 1 ) + (ψ 1 , φ 2 ) + (Bψ 1 , φ 2 ) .
With use of φ 2 = λφ 1 with λ purely imaginary and the fact that B is real and diagonal one rewrites the above equation to now integrating by parts we obtain Denoting the vectors of functional values and the outgoing derivatives at the vertices by Ψ and Ψ , respectively, and choosing Ψ = Φ, the above condition implies Ψ * Ψ − Ψ * Ψ = 0 for all Ψ, where * stands for hermitian conjugation. It is straightforward to check that this is equivalent to Ψ+iΨ = Ψ−iΨ , leading to the general coupling condition for any N ×N unitary matrix U and where I is the N ×N identity matrix. Let us note that the more general choice Ψ ± ilΨ instead of Ψ ± iΨ with any real l is possible; however, this leads to the condition (U l − I)Ψ + il(U l + I)Ψ = 0 and a simple transformation between U l and U shows that this equation is, in fact, equivalent to (5). The reverse implication that any pair of vectors Φ and Ψ satisfying the coupling condition (5), also satisfies (4), follows from properties of unitary matrices.
In agreement with the notation for quantum graphs we shall denote the coupling for which functional values are equal at each vertex and the sum of outward derivatives vanishes by standard coupling (the terms Kirchhoff's, Neumann or free coupling may also be found in the literature). The corresponding unitary coupling matrix is U = 2/d J − I, with d being the degree of the given vertex, I the d × d identity matrix and J the d × d matrix with all entries equal to 1.

The secular equation and asymptotic properties of eigenvalues and eigenfunctions
In this section we establish the asymptotic properties of the fundamental system of solutions and eigenvalues of H. Let us first consider a graph with all edges of length one. The first step uses the results of [BF09] for the construction of the fundamental system of solutions on a segment. We do not repeat the whole construction and instead present Lemma 3.1 from [BF09] on a fundamental system on the interval [0, 1].
Lemma 3.1 ( [BF09]). Let a ∈ C m+1 [0, 1] and b ∈ C m [0, 1]. Then there exist two linearly independent solutions u ± (x, λ) of equation (3) satisfying the initial condition u ± (0, λ) = 1 having the asymptotics and φ (±) The second step is to find a secular equation for a graph with edges of length one with coupling given by (5). We construct a flower-like model which is similar to the case studied for quantum graphs [Kuc04,EL10,DEL10]. The main idea, which we shall now describe, is very simple. For a general graph with N edges of unit length, one considers a one-vertex model with N loops also of unit length; the coupling matrix U is, in a suitably chosen basis, block diagonal with blocks corresponding to the vertex coupling matrices. The Hamiltonian on the original graph is unitarily equivalent to the Hamiltonian on the flower-like graph and hence every such graph can be described by this model. On each of the edges of the graph we take linear combinations of solutions of the form (6) as described below; the indices j denoting a particular edge are added where needed. We find Substituting this into (5) we obtain with the matrices M 1 and M 2 being block diagonal with blocks respectively. The secular equation is hence given by det [(U −I)M 1 (λ)+i(U + I)M 2 (λ)] = 0. Eigenfunctions corresponding to particular eigenvalues can be constructed from the eigenvector of the matrix in (8) corresponding to the zero eigenvalue. Another approach to constructing the secular equation is given in Section 5. Now we find the first term of the secular equation yielding the high frequency asymptotics, i.e. Im λ → ∞. Since u j± (1, λ) = e ±(λ+āj ) (1 + O(1/λ)) and the real part of the eigenvalues is bounded (see Corollary 4.2 below), we assume the following asymptotics for each sequence of eigenvalues λ ns = 2πin + c  The number of distinct high frequency abscissas of H will be called its abscissa count.
Let n + and n − denote respectively the number of +1's and −1's in the spectrum of U , and write with V being a unitary matrix and I k being the k × k identity matrix. The leading term of the high frequency asymptotics of the secular equation is given by with the diagonal blocks being successively n − × n − , n + × n + and (2N − n − − n + ) × (2N − n − − n + ) matrices. M 3 and M 4 consist in the basis given by V of blocks 1 1 e c0+āj e −c0−āj and 1 −1 −e c0+āj e −c0−āj , respectively. Note that there is a zero instead of D −I in the lower-right block of the matrix which multiplies M 3 , because only the first n − rows of the first matrix contribute to the term by n 2N −n− .
We shall now generalize the above results to graphs with edges of commensurate lengths.
Lemma 3.3. Let λ be an eigenvalue of a graph Γ with damping coefficients a j (x), potentials b j (x), all edges of length one and the coupling given by U . Then λ/l 0 is eigenvalue of the same graph with damping coefficients a j (x)/l 0 , potentials b j (x)/l 2 0 , all edges of length l 0 and coupling given by Proof. The wavefunction component on j-th edge of the first graph satisfies Substituting y = x/l 0 and rewriting the damped wave equation from which one finds the eigenvalues, damping and potential. Since with C being regular square matrix and hence one finds corresponding coupling matrix U l0 .
If the graph edges have different lengths, but all of them equal to multiples of l 0 , the above lemma can be used if we introduce additional vertices of degree 2.
The above construction of the first term of the secular equation is summarized in the following theorem which, in agreement with the one edge result in [CZ94,CFNS], states that for finding high frequency abscissas only the average of the damping coefficient on each edge is important.
Theorem 3.4. Let Γ be a graph with N edges of commensurate lengths l j = m j l 0 , m j ∈ N, j = 1, . . . , N , with the coupling conditions given by the coupling matrix U . Let the damping functions a j (x) and b j (x) be bounded and continuous on each edge. Let λ n be eigenvalues of the corresponding problem (3) and µ n eigenvalues for a j and b j replaced by their averages. Then the constant terms c 0 in the asymptotic expansion of λ n coincide with the corresponding constant terms in the asymptotic expansion of µ n .

Location of high frequency abscissas
We begin with a simple result stating that the real part of an eigenvalue is given by minus the average of the damping coefficients on each edge with weights taken as L 2 norms of the wavefunction components. A similar claim can be stated also for the damped wave equation on manifolds (see e.g. [AL03,Sjö00]).
Theorem 4.1. Let us consider a damped wave equation on a graph with N edges of lengths l j , bounded damping coefficients a j (x) and potentials b j (x), and the coupling conditions given by (5). If λ is an eigenvalue of H with nontrivial imaginary part Im (λ) = 0, then its real part satisfies where u j (x) denotes the corresponding wavefunction components.
Proof. Multiplying (3) byū j (x) and integrating over the j-th edge yields Now we sum this term over all edges. We show that the contribution of the term given by coupling conditions is real. Let U be the coupling matrix of the corresponding flower-like graph with N edges having n − eigenvalues equal to −1, n + eigenvalues equal to 1, and let the diagonal matrix D = diag (e iϕ1 , . . . , e iϕ 2N −n − −n + ) contain the remaining eigenvalues. We write it in the form with V being unitary and denote V Ψ = (Ψ 1 , Ψ 2 , Ψ 3 ) T and by V Ψ = (Ψ 1 , Ψ 2 , Ψ 3 ) T vectors of functional values and outward derivatives at a given vertex written in a suitable basis. Trivially, the subspaces corresponding to eigenvalues ±1 do not contribute to the second term of the above equation. Hence we have with Ψ 3s being the components of Ψ 3 . For the imaginary part of (9) we have and hence the claim of the theorem follows. For the real part, using (10) one gets 2 behaves asymptotically as n 2 for the eigenvalues λ n with large imaginary part. For certain types of graphs with standard coupling, some of the high frequency abscissas have a simple explicit form.
Requiring the functional values to be zero at the vertices of the loop one finds λ j (λ) = nπi, the continuity of the derivatives leads to α j+1 − α j (−1) n = 0 and thus (−1) nN = 0. Hence for N even one has two sequences with c

Pseudo orbit expansion and another type of secular equation
In what follows we shall use the approach using vertex scattering matrices and its pseudo orbit expansion to construct the secular equation. We refer to the pioneering articles in the case of quantum graphs by Kottos and Smilansky [KS97] and Akkermans et al.
[ACD + 00], the recent publication by Band, Harrison and Joyner [BHJ12] or the paper by Bolte and Endres where the case of general coupling was worked out [BE09].
In the case of the damped wave equation similar techniques can be used and, in a similar way to the cited works above, the orbit expansion can help finding the secular equation. From the secular equation, one can then find the leading term of its high frequency asymptotics (i.e. the expansion in n). In this section this expansion is done in terms of scattering matrices. Furthermore, we show how it is possible to obtain particular coefficients in the first term of this expansion using the orbit expansion.
We shall first briefly describe the method of orbit expansion. Let us replace the graph Γ by the directed graph Γ 2 , where the j-th edge of Γ is replaced by two directed edges e j andê j , both of length l j . Using the Ansatz The interpretation of the incoming and outgoing waves for this choice corresponds toλ j as the positive square root of λ 2 +2a e λ−b e and k = −Im (λ). We define the vertex scattering matrix σ v (λ) by α out We further denote S(λ) = JΣ(λ).
The following proposition compares the vertex scattering matrix behaviour in 1/n expansion with the coupling matrix for this particular vertex.
where d is the vertex degree.
Proof. The coupling condition becomes with Λ 2 being the diagonal matrix with entries c 0 + a j + O(1/n). This leads to the relation (12) Using the expansion of σ(λ) = σ 0 + 1 n σ 1 + 1 n 2 σ 2 + O(1/n 3 ) in n one finds from the first two terms For simplicity, we study the whole problem in the basis in which U is diagonal. Equation The fact that σ 0 is unitary yields σ 2 0 = σ 3 0 = 0 and since all upper blocks of the second and third terms in (14) vanish, we have σ 1 0 = −I n− . Rewriting σ(λ) into the appropriate basis we obtain the claim.
We use the same terminology as in [BHJ12].
Definition 5.2. A periodic orbit is a closed trajectory on the graph Γ 2 . An irreducible pseudo orbitγ is a collection of periodic orbits where none of the directed bonds is contained more than once. Let mγ denote the number of periodic orbits inγ, Lγ = e∈γλ e l e where the sum is over all directed bonds inγ andλ e = √ λ 2 + 2a e λ − b e . The coefficients Aγ = Π γj ∈γ A γj with A γj given as multiplication of entries of S(λ) along the trajectory γ j . Proof. The proof is similar to that of Theorem 1 in [BHJ12], except that the role of exp(iklγ) is replaced by exp(−Lγ(λ)). Its main essence is that nontrivial contributions to the determinant of (11) are given by such permutations which correspond to irreducible pseudo orbits. Each pseudo orbit of length s corresponds to taking s entries of the matrix S(λ) and 2N − s entries of −I. Since the number of −1's comming from the entries of this last matrix cancels with a part of the sign contribution of the permutation coming from the first term, the sign of a given contribution is given by the number of periodic orbits inγ.
Theorem 5.4. Let Γ be a graph with given damping coefficients a j (x) and potentials b j (x). At a fixed vertex v we assume a general coupling given by the matrix U .
i) If −1 ∈ σ(U ) then the abscissa count and high frequency abscissas will not change if U is replaced by U N = I, i.e. we separate all edges with Neumann coupling. The cases considered include e.g. δ s -condition (U d+iα J − I) then the damping count and high frequency abscissas will not change if U is replaced by standard coupling (α = 0). We emphasize that the case α = ∞, i.e. fully separated case, is not included.

Proof.
i) According to the second part of proposition 5.1 the first term of the asymptotics of the vertex scattering matrix equals the identity matrix. Hence the first term of the secular equation is the same as in the case of Neumann coupling. It is straghtforward to verify that the δ s -coupling matrix does not have −1 in its spectrum.
ii) The δ-coupling matrix has one eigenvalue equal to d−iα d+iα and an eigenvalue −1 with multiplicity d − 1. Hence, according to proposition 5.1, the first term in the expansion of the vertex scattering matrix is the same as that in the expansion of the vertex scattering matrix for the case of the standard coupling.

Number of distinct high frequency abscissas
Our next aim is to bound the number of sequences of eigenvalues with different real parts. We give an upper bound on the number of these sequences for an equilateral graph with general coupling conditions. Lemma 6.1. Let Γ be an equilateral graph with N edges of the length 1. Let us assume a damped wave equation on Γ with damping and potential functions constant on each edge a j (x) ≡ a j , b j (x) ≡ b j and with general coupling given by (5) for a given unitary matrix U . Then there exist numbers n 0 ∈ N, c (s) 0 ∈ C, s = 1, . . . , 2N and c 1 ∈ R such that for every n ≥ n 0 all eigenvalues of H are within the following set where B(x 0 , r) denotes the circle in the complex plane with center x 0 and radius r.
Proof. As follows from the construction in Section 3, high n asymptotics of the eigenvalues are given by λ ns = 2πin + c (s) We choose c 1 as twice the maximum of modulae of the coefficients by the O(1/n) for particular sequences and n 0 large enough for the O(1/n 2 ) term to be smaller than c 1 /n. It remains to prove that there are at most 2N different coefficients c 0 . Since the first term of the n asymptotics of the secular equation uniquely determines the coefficients c 0 , one can use the pseudo orbit expansion with the first term of n asymptotics of vertex scattering matrices. Each pseudo orbitγ of length lγ corresponds to a term by y lγ . Since the directed graph corresponding to Γ has 2N edges, we obtain a polynomial equation in y = e c0 of 2N -th order, which has 2N roots uniquely determining the values of c (s) 0 . We shall now consider some particular cases for which it is possible to be more specific regarding the possible number of high frequency abscissas. In the first example, we show that for bipartite graphs with N edges there are at most N spectral abscissas. To obtain a lower bound on the number of high frequency abscissas we shall first prove the following technical lemma. Lemma 6.3. Let Γ be a tree graph with standard coupling and all edges of length one, Γ 2 the corresponding oriented graph. Let {e 1 , . . . , e 2N } be a set of edges on Γ,γ a pseudo orbit on {e 1 , . . . , e 2N ,ê 1 , . . . ,ê 2N } ⊂ Γ 2 and X be a vertex of Γ of degree d and let there be v edges emanating from X denoted by {e 1 , . . . , e v }. Let s 1 = 2 d − 1, s 2 = 2 d be on-diagonal and off-diagonal elements of the scattering matrix at X , respectively. For a particular pseudo orbitγ let Γ 3 (γ ) be a collection of all pseudo orbits which can be obtained fromγ by all possible changes at X . Then the coefficient in γ∈Γ3(γ ) Aγ(λ) corresponding to the vertex X is Proof. First, we prove by induction that if one sums up the contribution of all paths with no reflection coefficients at X one obtains (1 − v)s v 2 . Let us assume a star graph of v edges and its corresponding directed counterpart. Let us denote by g(v) the (−1) mγ multiple of the number of different pseudo orbitsγ on it which do not have any reflection at the central vertex and let us assume g(v) = 1 − v. This clearly holds true for v = 2, 3, since g(2) = −1 (pseudo orbit e 1ê2 e 2ê1 ) and g(3) = −2 (pseudo orbits e 1ê2 e 2ê3 e 3ê1 and e 1ê3 e 3ê2 e 2ê1 ). Furthermore, we prove that g(v + 1) = vg(v) − vg(v − 1). One considers the set of all pseudo orbits without reflection on v edges. The first term corresponds to adding the (v + 1)-th edge to one of the pseudo orbits (for each pseudo orbit on r edges one has r possibilities and hence g(v) is multiplied by v). The second term corresponds to a pseudo orbit on two edges -the (v + 1)-th one and one of the others and any possible combination of nonreflection pseudo orbits on the other edges, i.e. g(v − 1). By induction we As the second step we use this result to prove the lemma. Then the coefficient obtained from pseudo orbit expansion by X is given by where in the middle term the expressions v−1 −1 and v−1 v are taken to be zero.
We may now show that under certain conditions it is always possible to have at least a number of high frequency abscissas which is equal to the number of edges in the graph.
Lemma 6.4. Let Γ be a tree graph with N edges all with unit length, Robin coupling at the boundary and standard coupling otherwise. Let us suppose that all vertices have odd degree. Then there always exists such a damping for which the number of high frequency abscissas is greater than or equal to N .
Proof. Let us assume a given set of the edges of the graph Γ and all pseudo orbits which go through every edge of this set twice. Then the contribution of all these pseudo orbits cancels (see e.g [ACD + 00]) if and only if there is a vertex of Γ of degree d = 2v and the above set of edges contains v edges emanating from this vertex. This follows from the previous lemma, since d = 2 − 2 s = 2 + 2(v − 1) = 2v for s = − 1 v−1 . We will prove the lemma by explicitly constructing the damping function for which this maximum of the number of sequences is attained. The idea is to choose the damping in a way that its average on each edge differs significantly. In the virtue of theorem 3.4 we choose constant damping on each edge, i.e. 0 a N a N −1 · · · a 1 . The first term of the n expansion of the secular equation can be written as (for simplicity we omit n to the corresponding power).
where y = e 2c0 and the coefficients C j are given by the orbit expansion. Since there are no vertices of degree two and standard conditions are considered, none of them is trivial. For y being close to e −2a1 the last two terms are dominant, for y close to e −2a2 the last-but-one and last-but-two terms, etc. Hence for the roots of the previous polynomial equation of the N -th order we get which gives The above result can be generalized for all equilateral graphs without vertices of degree two. The proof is very similar. Let us choose the damping coefficients again as 0 a N a N −1 · · · a 1 . Then the difference of the term of the pseudo orbit expansion of the secular equation corresponding to 2j + 1 edges and the term corresponding to 2j or 2j + 2 edges is O(e −aj +aj+1+aj+2 ), since the term with odd number of edges must contain at least three edges only once. Hence the secular equation for a graph with loops can be viewed as a small perturbation of a tree graph for this choice of the damping. Each of the roots (15) gives rise to two roots (in general not with distinct real parts) (with z j = e c (j) 0 ) which are still far away from the other roots.
To describe the distribution of eigenvalues and compare it with the twodimensional results we define the following probability measure (see also e.g. [AL03,Sjö00]). Definition 6.5. Let I be an open interval in R and R > 0. Then we define the probability distribution µ R (I) by µ R (I) = #{λ : Re λ ∈ I, |Im λ| < R} #{λ : |Im λ| < R} .
The above results can be written in the language of µ R (I) in the following way.
Theorem 6.6. Let Γ be a graph with N edges of length one, with coupling given by (5)  Proof. The second claim follows from lemma 6.1. m cj is an integer since the difference of imaginary part of two subsequent eigenvalues in each sequence λ ns is asymptotically 2π. Hence each c (j) 0 corresponding to one root of polynomial equation of the 2N -th order gives a sequence of eigenvalues with the counting function N j (R) = R/2π + O(1) and therefore proving ii). The first claim is a straightforward consequence of ii), while the third and fourth claims are reformulations of lemmas 6.2 and 6.4, respectively.

Examples
Now, we present several simple examples to illustrate the asymptotic behaviour of high frequency eigenvalues.
Example 7.1. Two loops with different damping coefficients Let us study an example of a graph with six edges which does not have any boundary vertices, consists of two loops, each of them having three edges of length one (see figure 1). Let the damping coefficients a 1 and a 2 be different on each loop and let us assume standard coupling at each of the vertices. We use the Ansatz f j (x) = α j sinh (λ j (λ)x) + β j cosh (λ j (λ)x) on both loops of the graph with x = 0 at their centers. Then from continuity conditions in the central vertex one has and hence either α 1 = α 2 = 0, or sinh 3 2λ 1 (λ) = 0 or sinh 3 2λ 2 (λ) = 0. Under the first assumption (α 1 = α 2 = 0) one has for standard conditions which can be written as (λ 1 +λ 2 ) sinh 3(λ 1 +λ 2 ) 2 + (λ 1 −λ 2 ) sinh 3(λ 1 −λ 2 ) 2 = 0 .
Using the asymptotic expansion λ n = 2πin + c 0 + O 1 n one obtains 4πin e 6πin+ 3 2 (a1+a2+2c0) − e −6πin− 3 2 (a1+a2+2c0) + O(1) = 0 and hence 3(a 1 + a 2 + 2c 0 ) + 2πin = 0 , On the other hand, the equation sinh 3 2λ j (λ n ) = 0 leads to The location of the eigenvalues for a particular choice of the damping coefficients a 1 and a 2 is shown in figure 2, there are in general three sequences of eigenvalues with real parts given by the previous relation and (16): −a 1 , −a 2 and − a1+a2 2 . The eigenfunctions corresponding to the first two of them are supported at each of the loops, the third one is supported on both of them and is symmetric on each of the loops.

. One loop and an appendix
The graph in the second example consists of a loop of length 3 and one edge of length one attached to this loop (see figure 3). We assume Dirichlet condition at the boundary vertex and standard coupling at the central one.
Using the Ansatz f j (x) = α j sinhλ j x on each edge with x = 0 at the free end one obtains the secular equation The eigenvalue location for particular lengths of the edges and the choice a 1 = 3, a 2 = 4, a 3 = 5 are shown in figures 10-12. From this we see that by choosing different lengths for the edges it will be possible to increase the number of high frequency abscissas and make it larger than the value of 2N given by lemmas 6.1 and 6.2 for the case where all abscissas have equal lengths.