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On the heat content of compact quantum graphs

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The heat content of a compact metric graph with Dirichlet conditions is determined exactly, for all times, by its volume, its number of Dirichlet vertices, and a scattering-weighted sum over closed paths hitting the boundary.

desk verdict Solid new heat-content formula for quantum graphs, but the small-time asymptotic proof rests on a false path-count bound; fixable, worth refereeing. read the letter →

arxiv 2502.09461 v1 pith:E7IR2DRY submitted 2025-02-13 math.SP math.APmath.CO

Patrizio Bifulco , Delio Mugnolo This is my paper · ORCID
classification math.SPmath.APmath.CO MSC 34B4505C5035P1581Q35
keywords heatcontentquantumgraphsmetricDirichletboundaryconditionssmall-timeasymptoticskernelscatteringcoefficientssurgeryprinciples
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves an exact, all-time formula for the heat content of a compact metric graph with Dirichlet conditions on a chosen set of degree-one vertices. The formula expresses the heat content as the graph's total length, minus a boundary correction proportional to the number of Dirichlet vertices and to $\sqrt{t}$, plus a sum over closed paths that touch the Dirichlet set and are weighted by scattering coefficients. The formula turns the heat content into a combinatorial quantity that can be read off from the graph's geometry, and it yields a short-time expansion whose leading coefficients record the volume and the number of Dirichlet vertices. The paper also derives comparison principles under graph surgeries, such as cutting loops, attaching graphs, and mirroring graphs.

What carries the argument

The argument rests on a path-sum representation of the heat kernel: the heat kernel $p^{G;V_D}_t(x,y)$ is written as $(4\pi t)^{-1/2}$ times a sum over directed paths from $x$ to $y$, each weighted by a scattering coefficient $\alpha(\vec p)$ and by $e^{-\ell(\vec p)^2/(4t)}$. Two combinatorial lemmas, a decomposition lemma and a summation lemma for scattering coefficients of extended paths, together reduce the double integral over $x,y$ to a single sum over closed paths that start and end at the Dirichlet set. The other key object is the function $H(x)=\pi^{-1/2}e^{-x^2}-x\,\mathrm{erfc}(x)$, which arises from integrating the one-dimensional Gaussian kernel and controls the exact remainder in the heat content formula.

What would settle it

On an equilateral star graph with $d>3$ Dirichlet leaves, compute the heat content exactly from the eigenfunction expansion in the paper and evaluate the right-hand side of the path-sum formula (4.1) with paths enumerated up to large combinatorial length; the two must agree as the truncation grows. To test the asymptotic bound, divide the remainder $Q_t-|G|+2\sqrt{t}/\sqrt{\pi}\#V_D$ by $\sqrt{t}\,e^{-\ell_{\min}^2/(4t)}$ and check boundedness as $t\to0^+$ on the same star, where the paper's counting estimate fails.

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Extended reading notes

Core claim

The central claim is that the heat content $Q_t(G;V_D)$ of a compact finite metric graph with Dirichlet conditions on a nonempty set of degree-one vertices equals $$Q_t(G;V_D)=|G|-\frac{2\sqrt{t}}{\sqrt{\pi}}\#V_D+8\sqrt{t}\sum_{p}\$\alpha$(p)\,H\!\left(\frac{\ell(p)}{2\sqrt{t}}\right)$$ for all $t>0$, where the sum runs over undirected paths that start and end at (possibly different) Dirichlet vertices, $\alpha(p)\in[-1,1]$ is a product of scattering coefficients collected at the vertices the path traverses, and $H$ is a monotone, exponentially decaying function. The same formula is equivalent to a sum over directed paths with factor $4\sqrt{t}$. As a consequence, the small-time expansion begins $$Q_t(G;V_D)=|G|-\frac{2\sqrt{t}}{\sqrt{\pi}}\#V_D+O\!\left(\sqrt{t}\,$e^{{-\ell_{\min}}$^2/(4t)}\right)\quad(t\to0^+),$$ with $\ell_{\min}$ the minimal edge length. The paper further states a Caccioppoli-type limit in which the heat flowing out of a subregion identifies the number of boundary points, and a set of surgery principles that compare heat contents of different graphs.

Load-bearing premise

The proof of the small-time asymptotics relies on the estimate that the number of directed paths of combinatorial length $n$ that start and end at the Dirichlet set is at most $2(d_{\max})^{n-1}$, where $d_{\max}$ is the maximal vertex degree; this estimate is false for an equilateral star with more than three Dirichlet leaves, so the remainder bound as written does not follow for such graphs.

Editorial extensions

If this is right

  • The heat content at any positive time can be computed from the graph's volume, the number of Dirichlet vertices, and the scattering-weighted set of closed paths hitting the boundary, without solving the heat equation.
  • As $t\to0^+$, $Q_t(G;V_D)=|G|-\frac{2}{\sqrt{\pi}}\#V_D\sqrt{t}+O\!\left(\sqrt{t}\,e^{-\ell_{\min}^2/(4t)}\right)$, so the first two coefficients of the short-time expansion are purely geometric.
  • For a closed connected subset $H$ of $G\setminus V_D$ whose boundary avoids vertices of degree at least three, the rescaled heat flowing from $H$ into its complement tends to $\#\partial H$ as $t\to0^+$.
  • Cutting a loop at its midpoint leaves the heat content unchanged at all times; mirroring a graph $m$ times multiplies the heat content by $m$; attaching a pendant graph increases the heat content; and lengthening an edge increases the heat content for small times.
  • Scaling all edge lengths by $s>0$ gives $Q_t(sG;V_D)=s\,Q_{t/s^2}(G;V_D)$ for all $t>0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exponentially decaying factor in the path sum suggests a practical numerical scheme: truncate the sum at a path-length cutoff and use the $H$ decay to control the error, with convergence expected to be rapid at moderate times.
  • The Hadamard-type derivative with respect to edge length, combined with the scaling law, could yield monotonicity or isoperimetric-type statements for the heat content under geometric variations, in the spirit of known results for the torsional rigidity.
  • The failure of the path-counting estimate on stars with many Dirichlet leaves indicates that a corrected proof of the small-time remainder should replace $d_{\max}$ by a factor involving $\#V_D$; the asymptotic statement itself may still be true.
  • Because the derivation uses only a path decomposition of the heat kernel plus the Gaussian kernel on $\mathbb{R}$, the same integration procedure should extend to Schrödinger operators or to other vertex conditions with adapted scattering coefficients.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies the heat content Q_t(G;V_D) of a compact finite metric graph with Dirichlet conditions imposed on a nonempty set V_D of degree-one vertices. The main result, Theorem 4.1, is an exact all-times formula expressing Q_t(G;V_D) as |G| - (2√t/√π)#V_D plus a scattering-weighted path sum over closed directed/undirected paths that start and end in V_D. The authors use this formula to derive a short-time asymptotic expansion, a Caccioppoli-type description of the perimeter of a subgraph, several surgery and comparison principles, and a Hadamard-type formula for variations of edge lengths. The proof of the main formula proceeds by integrating a Roth/KPS path-sum representation of the heat kernel and then simplifying the resulting sums through combinatorial lemmas in Section 3.

Significance. The exact heat-content formula is a significant contribution if it is correct: it gives a fully combinatorial description of the heat content at all times, with no fitted parameters, and it yields an exponentially small remainder in the short-time limit, which is stronger than the polynomial remainders typical in the manifold setting. The surgery principles, especially the mirroring formula of Theorem 6.8 and the loop-cut invariance of Proposition 6.4, are likely to be useful for spectral-geometric comparisons of quantum graphs. The paper also contains explicit worked examples, including intervals, stars, lassos, and pumpkin chains, which help to make the path-sum formalism concrete. The central all-time formula of Theorem 4.1 appears derivationally sound; the principal defect is a false path-counting estimate used in the proof of the small-time asymptotics and of Corollary 6.12.

major comments (2)
  1. [§5, Eq. (5.5)] Equation (5.5) is false. It claims #(P_n(G) ∩ P_{V_D}(G)) ≤ 2(d_max)^{n-1} for every n, but the bound does not depend on #V_D. Let G be an equilateral star with d ≥ 4 leaves, all leaves lying in V_D and the centre in V_N; Assumption 2.1 holds. Then every ordered pair of leaves (v_i,v_j) gives a directed path v_i → centre → v_j of combinatorial length 2, so #(P_2(G) ∩ P_{V_D}(G)) = d^2, while the claimed upper bound is 2d. The extra factor 2 is also unnecessary, since P_n(G) already consists of directed paths. Thus the estimate (5.6) is not established by the argument given.
  2. [§5, Eq. (5.6)–(5.8) and §6.4, Eq. (6.14)] Because (5.5) is the only control on the number of paths used to bound the remainder L_t(G;V_D) in (5.3), the small-time bound (5.1) and the O-term (5.2) of Theorem 5.1 are not proved as written. The same false estimate is invoked in (6.14) in the proof of Corollary 6.12, so that result also lacks a valid proof. This is a repairable proof gap rather than a disproof of the asymptotic claim: replacing (5.5) by the valid bound #(P_n(G) ∩ P_{V_D}(G)) ≤ #V_D (d_max)^{n-1} gives an additional factor #V_D in (5.6) and (6.14) but preserves the exponential order, so the stated O(√t e^{-ℓ_min^2/(4t)}) remainder is likely correct. I ask the authors to correct (5.5) and to rerun the estimates in Theorem 5.1 and Corollary 6.12 accordingly.
minor comments (6)
  1. [Title] The displayed title contains broken spacing in 'HEA T CONTENT' and 'COMP ACT'; this should be corrected in the final version.
  2. [Theorem 4.1 and Eq. (1.3)] The statement of Theorem 4.1 in (4.32) is in terms of directed paths with a factor 4√t, while Theorem A and (1.3) use undirected paths with a factor 8√t. This is consistent with the two orientations of each undirected path, but the equivalence should be stated explicitly when the theorem is announced.
  3. [Example 4.10] In the displayed computation for the lasso graph, the quadruple sum over n,k,ℓ,j and the condition n+k≤m+1 are introduced without explaining how m and the four counters correspond to the path statistics R_{v_D}, R^{(1)}_{v_N}, R^{(2)}_{v_N}, T^{(1)}_{v_N}, T^{(2)}_{v_N}; the example would be much clearer if this dictionary were spelled out.
  4. [Corollary 6.3, proof] The proof says that imposing an additional Dirichlet condition 'raises the heat content by Proposition 6.2', but Proposition 6.2 states the opposite inequality, Q_t(G;V_D∪{v_0}) ≤ Q_t(G;V_D). The wording should be corrected and the three steps of the surgery argument should be reordered more explicitly.
  5. [Theorem 5.5, Eq. (5.15)] The bound (5.15) counts directed topological paths by assigning at most d_max choices at each vertex, but the role of the parameter n and the extra factor d_max^{n+1} are not precisely defined. A short counting lemma analogous to a corrected (5.5) would make this estimate rigorous and easier to verify.
  6. [Remark 5.2] In the displayed heat-kernel asymptotics, the notation dist(x,V) is used but the distance to the vertex set V is not defined until later in the same remark; please define it before first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.1 derives heat content from the external Roth/KPS path-sum formula, and no fitted parameter or self-referential input forces the claimed result.

full rationale

The central derivation is a direct integration of Proposition 4.2, the Roth/Kostrykin–Potthoff–Schrader path-sum formula, which is cited to external prior work and does not assume the target heat-content formula. The heat-content formula (4.1) then follows by algebraic identities, Gaussian integral evaluations, and scattering-coefficient summation lemmas (Lemmas 3.6, 3.7, 4.9); the derivation does not fit any parameter or normalize by the quantity being predicted. The small-time asymptotics (Theorem 5.1) is deduced from (4.1) by bounding the path remainder with (5.5); the bound in (5.5) is in fact false as stated for stars with d>3 Dirichlet leaves, since an equilateral star has #(P_2(G)∩P_VD(G))=d^2 while the claimed bound gives 2d^{2-1}=2d. Consequently the estimates (5.6) and (5.8), hence Theorem 5.1 and Corollary 6.12, are not proved as written; this is a correctness defect in a derivative asymptotic result, not a circularity, because the asymptotic claim is derived from the external path representation rather than assumed as an input. The self-citations in the paper ([13] for Lipschitz continuity of the heat kernel, [14] for the Feynman–Kac comparison used in Proposition 6.5) are auxiliary and are not invoked to define or force the main formula; Proposition 6.5 does omit its proof and defers to [14], but that is a verification gap in a secondary comparison principle, not a circular reduction. Overall the derivation chain is self-contained against the external path-sum result, so no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central formula rests on the external Roth/KPS path-sum heat kernel representation and on the degree-one Dirichlet vertex assumption. No free parameters are fitted. One additional path-counting bound used for the small-time asymptotics is asserted without proof and is false as written, which is the main honesty concern in the ledger.

assumptions (4)
  • standard math The heat kernel of Delta_{G;V_D} admits the Roth/KPS path-sum representation (Proposition 4.2), summed over directed paths with scattering coefficients alpha(p) and Gaussian weights exp(-ell(p)^2/(4t)).
    Quoted from [48] and [28, Section 3.4]; the entire derivation of Theorem 4.1 starts from this representation.
  • domain assumption Assumption 2.1: G is compact and finite, V_D is a nonempty subset of degree-one vertices, G \ V_D is connected, and G has no degree-2 vertices.
    Restricts the class of graphs; the scattering formalism in Definition 3.2 is tailored to Dirichlet vertices of degree one.
  • domain assumption Term-by-term integration and summation interchange over the path sums in Proposition 4.8 and Theorem 4.1 are legitimate.
    The paper states convergence in Proposition 4.8(i) and proceeds, but does not give a fully detailed justification of every interchange; if this failed, Theorem 4.1 would not follow.
  • ad hoc to paper The number of directed paths from V_D to V_D of combinatorial length n is at most 2 (dmax)^{n-1}, as asserted in (5.5).
    Used in the proof of Theorem 5.1 and Corollary 6.12; this estimate is false for star graphs with many Dirichlet leaves.

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Pith. "Pith review of On the heat content of compact quantum graphs." pith.science (2026). https://pith.science/paper/E7IR2DRY

@misc{pith2026250209461,
  author       = {Pith},
  title        = {Pith review of: On the heat content of compact quantum graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7IR2DRY}},
  note         = {Machine review of arXiv:2502.09461}
}
abstract

We study the heat content for Laplacians on compact, finite metric graphs with Dirichlet conditions imposed at the "boundary" (i.e., a given set of vertices). We prove a closed formula of combinatorial flavour, as it is expressed as a sum over all closed orbits hitting the boundary. Our approach delivers a small-time asymptotic expansion that delivers information on crucial geometric quantities of the metric graph, much in the spirit of the celebrated corresponding result for manifolds due to Gilkey-van den Berg; but unlike other known formulae based on different methods, ours holds for all times $t>0$ and it displays stronger decay rate in the short time limit. Furthermore, we prove new surgery principles for the heat content and use them to derive comparison principles for the heat content between metric graphs of different topology.

Figures

Figures reproduced from arXiv: 2502.09461 by the authors.

Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 2.1
Figure 2.1. The graph G with one Dirichlet condition at 0 on the left and two Dirichlet conditions at 0 and ℓ on the right (Dirichlet conditions are imposed at the white vertices). for k ∈ N, x ∈ [0, ℓ], and thus, for the heat content, we obtain – using (2.4) – that Qt([0, ℓ]; VD) = 8ℓ π 2 X∞ k=0 e −t [PITH_FULL_IMAGE:figures/full_fig_p011_2_1.png] view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figures from the paper (34 more)
Figure 2.2
Figure 2.2. Figure 2.2: The profiles of Qt([0, ℓ]; VD) for ℓ = 3 and #VD = 1 (left), #VD = 2 (right)... (2) Let G ≃ Sn(ℓ) = Sn be the equilateral star graph on n + 1 vertices with Dirichlet conditions at all outer vertices, i.e., #VD = n, and with same edge length 0 < ℓ < ∞, see [PITH_FULL…
Figure 2.3
Figure 2.3. Figure 2.3: and the profiles of √ π 2 √ t (Qt([0, ℓ]; VD) − ℓ), again for ℓ = 3 and #VD = 1 (left), #VD = 2 (right) [PITH_FULL_IMAGE:figures/full_fig_p012_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: The star graph S5 with 5 edges and Dirichlet conditions imposed at all white vertices. A tedious but elementary computation eventually yields the explicit expansion Qt(Sn; VD) = 8|Sn| π 2 X∞ k=0 e −t [PITH_FULL_IMAGE:figures/full_fig_p012_2_4.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png]
Figure 2.5
Figure 2.5. Figure 2.5: The profiles of Qt(Sn; VD) (left) and √ π 2 √ t (Qt(S3; VD) − 3ℓ) (right) for ℓ = 1 and n = 3 Definition 3.1 (Directed paths). Two bonds e⃗1 and e⃗2 are called consecutive if ∂ +(e⃗1) = ∂ −(e⃗2). Let n ∈ N0 and v,w ∈ V. A directed path p⃗ from v to w (which is called…
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png]
Figure 3.1
Figure 3.1. Figure 3.1: A star graph S3 with 3 edges and a Dirichlet condition together with a nontrivial directed path p⃗ = (v, e⃗1, e⃗2, e⃗3, e⃗4,w) (in blue) and corresponding scattering coefficient α(p⃗) = ( 2 3 − 1) · (−1) · 2 3 = 2 9 . For v,w ∈ V, we denote by Pm(v,w) the set of path…
Figure 3.2
Figure 3.2. Figure 3.2: A nontrivial directed path p⃗ = [PITH_FULL_IMAGE:figures/full_fig_p015_3_2.png]
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png]
Figure 3.3
Figure 3.3. Figure 3.3: The star graph S3 together with the final bond e⃗+(p⃗) of a path p⃗ ending at v0 on the left and the corresponding final bonds e⃗i , i = 1, 2, 3, of all paths in ⟨p⃗⟩+ on the right. Likewise, if p⃗ ∈ P→W(G), i.e. p⃗ is a path ending at an outer vertex vi ∈ W, i = 1, …
Figure 3.4
Figure 3.4. Figure 3.4: The star graph S3 together with the final bond e⃗+(p⃗) of a path p⃗ ending at W on the left and the corresponding final bond e⃗+(q⃗) of the single path q⃗ ∈ ⟨p⃗⟩+ on the right. Lemma 3.6 (Decomposition Lemma). Let W ⊂ V. Then PW→(G) = [ q⃗∈PW→(G) ⟨q⃗⟩+ ⊔  PW→(G) ∩ P…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: The profile of the function H in (4.9) Proof. Let x, y ∈ G. Given a path p⃗ = (x, e⃗1, . . . , e⃗n, y) ∈ P≥2(x, y) with n ≥ 2, we can find bonds e⃗, ⃗f ∈ B such that x lies on the corresponding edge e ∈ E and y lies on the corresponding edge f ∈ E and such that e⃗ an…
Figure 4.2
Figure 4.2. Figure 4.2: The path p⃗ = (x, e⃗1, . . . , e⃗n, y) and edges e ∋ x and f ∋ y such that ∂ +(e⃗) = ∂ +(e⃗1) and ∂ −( ⃗f) = ∂ −(e⃗n). Thus, we have a one-to-one correspondence between P≥2(x, y) and [ e⃗∈b(e), ⃗f∈b(f) n p⃗ ∈ P≥2(G) : e⃗−(p⃗) = e⃗, e⃗+(p⃗) = ⃗f o , (4.12) where every…
Figure 4
Figure 4. Figure 4: ) [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 4.3
Figure 4.3. Figure 4.3: A lasso graph with a Dirichlet condition at vD. Given p⃗ ∈ P({vD}), i.e., a directed path starting and ending at vD, we can count the number • Rp⃗ (vD) of times p⃗ touches vD; • R (1) p⃗ (vN) of times p⃗ arrives along e1, touches vN, and is reflected back into e1; • …
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]
Figure 6.1
Figure 6.1. Figure 6.1: A graph G with a closed subgraph H ⊂ G (blue). To begin with, we observe that ∆H;VD,H generates a positive strongly continuous semigroup (et∆ H;VD,H )t≥0, which is dominated by the original semigroup (et∆G;VD )t≥0 as the following proposition states. Proposition 6.1.…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 6.2
Figure 6.2. Figure 6.2: The lasso graph G from Example 4.10 (left) and a 3-star graph Ge arising from a midpoint loop cut in the sense of Proposition 6.4 through the edge e at v0. Proof. Let e +, e − denote the two pendant edges of Ge arising through the midpoint cut through e and denote by…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p036_6.png]
Figure 6.3
Figure 6.3. Figure 6.3: The lasso graph G (left) and the 3-star graph Ge arising from a midpoint loop cut in the sense of Proposition 6.4 through the edge e as in [PITH_FULL_IMAGE:figures/full_fig_p036_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: A directed path p⃗ = [PITH_FULL_IMAGE:figures/full_fig_p037_6_4.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p038_6.png]
Figure 6.5
Figure 6.5. Figure 6.5: below). v (i) w0 w (i) ev w ′ 0 we w ′ 1 ev w ′′ 0 we w ′′ 1 w ′′ 2 G ≃ Gi G (2)(V0) G (3)(V0) V0 [PITH_FULL_IMAGE:figures/full_fig_p038_6_5.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p039_6.png]
Figure 6.6
Figure 6.6. Figure 6.6 [PITH_FULL_IMAGE:figures/full_fig_p039_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: The lasso graph G from [PITH_FULL_IMAGE:figures/full_fig_p040_6_7.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p041_6.png]
Figure 6.8
Figure 6.8. Figure 6.8: A 3-star graph G together with a directed path p⃗ (left) with αG(p⃗) = β1 (i.e., αfG(p⃗) = 1 in this case) and the corresponding 3-fold mirrored graph G3({v0}) to￾gether with corresponding paths p⃗(1, 1), p⃗(1, 2), p⃗(1, 3) (right) following the same bonds as p⃗ poss…
Figure 6.9
Figure 6.9. Figure 6.9: The 3-star graph G from [PITH_FULL_IMAGE:figures/full_fig_p042_6_9.png]
Figure 6.10
Figure 6.10. Figure 6.10: The 3-star graph G from [PITH_FULL_IMAGE:figures/full_fig_p043_6_10.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p045_6.png]
Figure 6.11
Figure 6.11. Figure 6.11: A 3-regular pumpkin chain with h = 4 and #VD = 1. 6.4. A Hadamard-type formula. Given a graph G as in Assumption 2.1 and fix an edge e0 ∈ E. We consider its length-perturbed graph Gs for some s > 0 which is defined by the same topology as G but with edge lengths (ℓe…

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