This section is devoted to formulate and prove the following.

Theorem 4.1

(Heat content formula). The heat content satisfies

$$\begin \begin \mathcal _t(\mathcal ;_}})&= \vert \mathcal \vert - \frac}} \# _}}+ 4\sqrt \sum _ \alpha (\vec p) \, H\bigg ( \frac} \bigg ) \qquad \hbox t>0, \end \end$$

(4.1)

where the summation in (4.1) runs over all directed paths starting and ending at \(_}}\).

The proof of Theorem 4.1 will require a number of auxiliary results and will be completed at the end of this section.

To begin with, let us recall the path sum formula first obtained in [47] for \(_}}=\emptyset \) and, in the general case of possibly nonempty \(_}}\), in [26, Section 3.4].

Proposition 4.2

(Path sum formula). Let \(\mathcal \) be as in Assumption 2.1 and let \(\textsf,\textsf\in \textsf\). Then, the heat kernel \(p_t^;_}}}(\cdot ,\cdot )\) associated with the Dirichlet Laplacian \(\Delta ^;_}}}\) is given by

$$\begin p_t^;_}}}(x ,y)&= \frac} \sum _ \in \mathcal _(x,y)} \alpha (\vec ) \textrm^} \end$$

(4.2)

$$\begin&= \frac} \, \delta _,\textsf} \, \textrm^\,}}_\mathcal ( x, y)^2}} + \frac}\sum _ \in \mathcal _(x,y)} \alpha (\vec ) \textrm^} \end$$

(4.3)

for all \(t>0\) and all points \(x,y\in \mathcal \) that lie in the interior of edges \(\textsf,\textsf\), respectively, with uniformly convergent right-hand side on \(\mathcal \times \mathcal \) for fixed \(t>0\).

Remark 4.3

(1) Inserting artificial vertices of degree 2 does not change \(\Delta ^;_}}}\) or its heat kernel. Therefore, the subclasses \(\mathcal P_(x,y)\) and \(\mathcal P_(x,y)\) of directed paths between x, y are well-defined.

(2) Let \(y \in _}}\). As observed in [12, Proposition 2.1], (4.3) also holds whenever \(x\in _}}\). However, (4.3) may be wrong whenever \(x\in _}}\), as can be seen in the case of an interval \(\mathcal \simeq [0,\ell ]\) with mixed boundary conditions (Dirichlet at 0, Neumann at \(\ell \)): indeed, \(p_t^;_}}}(x,y)=0\) since \(p_t^;_}}}(\cdot ,y)\) lies in the domain of \(\Delta ^;_}}}\), while for the right-hand side of (4.2) there holds

$$ \frac} \sum _^\infty (-1)^\textrm^} \ge \frac} \bigg ( \textrm^} - \textrm^} \bigg ) > 0 = p_t^;_}}}(x,y). $$

The first addend of (4.3) does not depend on the scattering coefficients and thus does not encode the topology of the graph \(\mathcal \). This motivates us to introduce the following notions as both terms appearing in (4.3) belong to \(L^\infty (\mathcal \times \mathcal )\) for all \(t>0\).

Definition 4.4

The non-topological part of the heat content is

$$ \mathcal _t^}(\mathcal ):= \frac} \sum _,\textsf\in \textsf} \int _0^} \int _0^} \delta _,\textsf} \, \textrm^\,}}_\mathcal ( x, y)^2}} \,\textrmy \,\textrmx, \qquad t>0, $$

whereas the topological part is

$$ \mathcal _t^}(\mathcal ;_}}):= \frac} \int _\mathcal \int _\mathcal \sum _ \in \mathcal _(x,y)} \alpha (\vec ) \textrm^} \,\textrmy \,\textrmx, \qquad t >0. $$

(We stress that \(\mathcal _t^}(\mathcal )\) does indeed not depend on \(_}}\)!) From now on and throughout this section, we suppose the underlying graph \(\mathcal \) as well as the set \(_}}\) to be fixed, and write \(\mathcal _t\), \(\mathcal _t^}\) and \(\mathcal _t^}\) rather than \(\mathcal _t(\mathcal ;_}}), \mathcal _t^}(\mathcal )\) and \(\mathcal _t^}(\mathcal ;_}})\), respectively, to simplify the notation.

By construction we have that \(\mathcal _t = \mathcal _t^}+ \mathcal _t^}\), and it will be beneficial for us to study both parts separately: as the name however suggests, the non-topological part is just given by integration on each edge, separately: a first immediate observation is that the non-topological part of the heat content can be written as

$$\begin \mathcal _t^}= \frac}\sum _\in \textsf} \int _0^} \int _0^} \textrm^} \,\textrmy \,\textrmx \qquad }\, t>0. \end$$

(4.4)

Let us present a further simple representation based on the complementary error function \(\textrm \in C^\infty (\mathbb )\) defined by

$$ \textrm(x):= \frac}\int _x^\infty \textrm^ \,\textrms, \qquad x \in \mathbb , $$

cf. also [1, 26] (note that \(\frac}x} \textrm(x) = -\frac} \textrm^\) for any \(x \in \mathbb \)).

Lemma 4.5

There holds

$$\begin \mathcal _t^}= \vert \mathcal \vert + \frac} \sqrt \bigg ( \sum _\in \textsf} \textrm^^2}} - \# \textsf\bigg ) - \sum _\in \textsf} \ell _\textsf\textrm\bigg ( \frac}} \bigg ) \qquad }\, t>0. \end$$

(4.5)

Proof

First, for any \(\textsf\in \textsf\) we have

$$\begin \int _0^} \int _0^} \textrm^} \,\textrmy \,\textrmx&= 4t\Big ( \textrm^^2}} - 1 \Big ) + 2 \sqrt \ell _\textsf\bigg (1 - \textrm \bigg ( \frac}} \bigg )\bigg ). \end$$

(4.6)

Indeed, defining \(f(\ell ,x):= \int _0^\ell \textrm^ \,\textrmy\) for \(\ell \ge 0\) and \(x \in [0,\ell ]\), we observe according to Leibniz integral rule that

$$\begin \begin \frac}\ell } \int _0^\ell \int _0^\ell \textrm^ \,\textrmy \,\textrmx&= f(\ell ,\ell ) + \int _0^\ell \frac}\ell } f(\ell ,x) \textrmx \\ &= \int _0^\ell \textrm^ \,\textrmy + \int _0^\ell \textrm^ \,\textrmx = \sqrt (1-\textrm(\ell )) \end \end$$

(4.7)

and integrating both sides of (4.7) yields

$$\begin \begin \int _0^\ell \int _0^\ell \textrm^ \,\textrmy \,\textrmx&= \sqrt \int _0^\ell (1-\textrm(s)) \,\textrms \\ &= \sqrt\ell (1-\textrm(\ell )) + \textrm^ - 1, \end \qquad }\, \ell \in \mathbb _+, \end$$

which eventually yields (4.6) through substitution. This shows the claim. \(\square \)

The expression in (4.5) can also be rewritten as

$$\begin \begin \mathcal _t^}&= \vert \mathcal \vert - \frac}}\# \textsf+ 2\sqrt \sum _\in \textsf} \Bigg ( \frac} \textrm^^2}} - \frac}} \textrm\bigg (\frac}} \bigg ) \Bigg ) \\ &= \vert \mathcal \vert - \frac}}\# \textsf+ 2\sqrt \sum _\in \textsf} H \bigg ( \frac}} \bigg ) \ \end, \qquad t > 0, \end$$

(4.8)

where

$$\begin H(x) := \frac} - \int _0^x \textrm(s)\,\textrms = \frac}\textrm^ - x\textrm(x), \qquad x \in \mathbb . \end$$

(4.9)

Let us summarize a few basic properties of H, which we plot in Fig. 10.

Lemma 4.6

The function H belongs to \(C^\infty (\mathbb )\) and is strictly convex. Furthermore, \(H(\mathbb ) \subset (0,\infty )\).

Proof

A direct computation shows that

$$ H'(x) = -\textrm(x)<0,\quad H''(x) = \frac^}}>0\qquad \hbox x \in \mathbb , $$

hence \(H\in C^\infty (\mathbb )\) and H is strictly decreasing and convex. Moreover,

$$\begin \textrm(x) = \frac} \int _x^\infty \textrm^ \,\textrmt < \fracx} \int _x^\infty 2t\textrm^ \,\textrmt = \frac^}x}\qquad \hbox x \in \mathbb _+, \end$$

(4.10)

whence \(H(\mathbb _+) \subset (0,\infty )\). Moreover, it is an immediate observation that \(H(\mathbb \setminus \mathbb _+) \subset (0,\infty )\). \(\square \)

Fig. 10

The profile of the function H in (4.9)

Showing that the topological part of the heat content, too, can be represented in terms of H will be the next major step on the way to prove Theorem 4.1. To this aim, one needs to establish a suitable relation between \(\mathcal _(x,y)\) (the set of directed paths between x and y of combinatorial length at least 2, see (3.4)) and \(\mathcal _(\mathcal )\) (the set of directed paths on \(\mathcal \) of combinatorial length at least 2, see (3.5)), in order to swap the integral with the summation over all directed paths in \(\mathcal _(x,y)\).

Lemma 4.7

One has

$$\begin \mathcal _t^}= \frac} \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) \int _0^_-(\vec )}} \int _0^_+(\vec )}} \textrm^_\pm ) + y)^2}} \,\textrmy \,\textrmx \qquad }\, t>0. \end$$

(4.11)

.

Proof

Let \(x,y \in \mathcal \). Given a path \(\vec =(x,\vec }_1,\dots ,\vec }_n,y) \in \mathcal _(x,y)\) with \(n \ge 2\), we can find bonds \(\vec }, \vec } \in \textsf\) such that x lies on the corresponding edge \(\textsf\in \textsf\) and y lies on the corresponding edge \(\textsf\in \textsf\) and such that \(\vec }\) and \(\vec }\) are directed into the same direction as \(\vec }_1\) and \(\vec }_n\) in the sense that \(\partial ^+(\vec }) = \partial ^+(\vec }_1)\) and \(\partial ^-(\vec }) = \partial ^-(\vec }_n)\) (see Fig. 11).

Fig. 11

The path \(\vec = (x,\vec }_1,\dots ,\vec }_n,y)\) (following the dashed directed edges) and edges \(\textsf\ni x\) and \(\textsf\ni y\) with corresponding directed edges \(\vec }, \vec }\) (dashdotted lines) such that \(\partial ^+(\vec }) = \partial ^+(\vec }_1)\) and \(\partial ^-(\vec }) = \partial ^-(\vec }_n)\)

Thus, we have a one-to-one correspondence between \(\mathcal _(x,y)\) and

$$\begin \bigcup _} \in \textbf(\textsf), \vec } \in \textbf(\textsf)} \left\ \in \mathcal _(\mathcal ) \, : \, \vec }_-(\vec ) = \vec }, \, \vec }_+(\vec ) = \vec } \right\} , \end$$

(4.12)

where every path \(\vec =(x,\vec }_1,\dots ,\vec }_n,y) \in \mathcal _(x,y)\) corresponds to a path

$$ \vec (},\vec }}):= \big (\partial ^-(\vec }), \vec },\vec }_2,\dots ,\vec }_,\vec }, \partial ^-(\vec }) \big ) \in \mathcal (\mathcal ) $$

with \(\vec } \in \textbf(\textsf)\) and \(\vec } \in \textbf(\textsf)\) such that \(\partial ^+(\vec }) = \partial ^+( \vec }_1)\) and \(\partial ^-(\vec }) = \partial ^-(\vec }_n)\), as well as

$$\begin \alpha (\vec p) = \alpha \big (\vec (},\vec }}) \big ), \end$$

(4.13)

and

$$\begin \begin \ell (\vec p)&= \,}}(x,\partial ^+(}}_1)) + \ell \big (\vec (},\vec }})_\pm \big ) + \,}}(\partial ^-(}}_),y)\\&= \,}}(x,\partial ^+(}})) + \ell (\vec _\pm ) + \,}}(\partial ^-(}}),y), \end \end$$

(4.14)

as \(\vec (},\vec }})_\pm = \vec _\pm \) by construction. Moreover, one has

$$\begin&\,}}(x,\partial ^+(\vec })) = \ell _\textsf-x, & }\, \overset}\ \partial ^+(\vec }_1) , \\ x , & }\, \partial ^+(\vec }_1) \overset}, \end\right. } \quad } \nonumber \\&\quad \,}}(\partial ^-(\vec }),y) = \ell _\textsf- y, & }\, \overset}\ \partial ^-(\vec }_n) , \\ y , & }\, \partial ^-(\vec }_n) \overset}. \end\right. } \end$$

(4.15)

By appropriate substitution, one sees that

$$\begin&\int _0^ \int _0^ \textrm^} \,\textrmy \,\textrmx = \int _0^ \int _0^ \textrm^} \,\textrmy \,\textrmx \\ &\quad \qquad = \int _0^ \int _0^ \textrm^} \,\textrmy \,\textrmx = \int _0^ \int _0^ \textrm^} \,\textrmy \,\textrmx \end$$

for all \(\alpha ,\beta ,\ell \in \mathbb _+\): therefore,

$$\begin \int _0^} \int _0^} \textrm^\,}}(x,\partial ^+(\vec })) + \ell (\vec _\pm ) + \,}}(\partial ^-(\vec }),y))^2}} \,\textrmy \,\textrmx = \int _0^} \int _0^} \textrm^_\pm ) + y)^2}} \,\textrmy \,\textrmx, \end$$

(4.16)

for all \(\vec },\vec } \in \textsf\), all \(\vec p \in \mathcal _(\partial ^-(\vec }), \partial ^+(\vec }))\), and all \(t > 0\). Combining now (4.13), (4.14) and (4.16), this eventually leads to

$$\begin \mathcal _t^}&= \frac} \sum _,\textsf\in \textsf} \int _0^} \int _0^} \sum _ \in \mathcal _(x,y)} \alpha (\vec p) \textrm^)^2}} \,\textrmy \,\textrmx \\ &= \frac} \sum _,\textsf\in \textsf} \int _0^} \int _0^} \sum _} \in \textbf(\textsf), \, \vec } \in \textbf(\textsf)}\sum _}_-(\vec ) = \vec }, \, \vec }_+(\vec ) = \vec }}\limits ^ \in \mathcal _(\mathcal )}}} \alpha (\vec p) \textrm^\,}}(x,\partial ^+(\vec })) + \ell (\vec _\pm )+\,}}(\partial ^-(\vec }),y))^2}} \,\textrmy \,\textrmx \\ &= \frac} \sum _,\textsf\in \textsf} \sum _} \in \textbf(\textsf), \, \vec } \in \textbf(\textsf)}\sum _}_-(\vec ) = \vec }, \, \vec }_+(\vec ) = \vec }}\limits ^ \in \mathcal _(\mathcal )}}} \alpha (\vec p) \int _0^} \int _0^} \textrm^_\pm )+y)^2}} \,\textrmy \,\textrmx \\ &= \frac} \sum _}, \vec } \in \textsf} \sum _}_-(\vec ) = \vec }, \, \vec }_+(\vec ) = \vec }}\limits ^ \in \mathcal _(\mathcal )}}} \alpha (\vec p) \int _0^} \int _0^} \textrm^_\pm )+y)^2}} \,\textrmy \,\textrmx \\ &= \frac} \sum _ \in \mathcal _(\mathcal )} \alpha (\vec p) \int _0^_-(\vec )}} \int _0^_+(\vec )}} \textrm^_\pm )+y)^2}} \,\textrmy \,\textrmx: \end$$

this completes the proof. \(\square \)

Using Leibniz’ integral rule to compute the terms of the form \( \int _0^\alpha \int _0^\beta \textrm^} \,\textrmy \,\textrmx \) for \(\alpha ,\beta ,\ell \in \mathbb _+\), in a similar way to the proof of Lemma 4.5, we obtain the following first combinatorial expression for the heat content which also involves the function H.

Proposition 4.8

Under Assumption 2.1 the following assertions hold.

(i) For \(t>0\) each of the series

$$\begin & \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) H\bigg ( \frac)}} \bigg ), \,\, \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) H\bigg ( \frac_-)}} \bigg ),\,\, \\ & \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) H\bigg ( \frac_+)}} \bigg ),\,\, \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) H\bigg ( \frac_\pm )}} \bigg ) \end$$

converges.

(ii) The topological part of the heat content satisfies

$$\begin \mathcal _t^}= \sqrt \sum _ \in \mathcal _(\mathcal )} \alpha (\vec ) \left( H\bigg ( \frac)}} \bigg ) - H\bigg ( \frac_-)}} \bigg ) - H\bigg ( \frac_+)}} \bigg ) + H\bigg ( \frac_\pm )}} \bigg ) \right) \end$$

(4.17)

for all \(t>0\).

In particular, Proposition 4.8 shows that the heat content itself satisfies

$$\begin \begin \mathcal _t&= \vert \mathcal \vert + \frac}} \# \textsf- 2\sqrt \sum _\in \textsf} H\bigg ( \frac}} \bigg ) \\ &\qquad +\sqrt \sum _ \in \mathcal (\mathcal )} \alpha (\vec ) \left( H\bigg ( \frac)}} \bigg ) - H\bigg ( \frac_-)}} \bigg ) - H\bigg ( \frac_+)}} \bigg ) + H\bigg ( \frac_\pm )}} \bigg ) \right) \\ &= \vert \mathcal \vert + \frac}} \# \textsf- 2\sqrt \sum _\in \textsf} H\bigg ( \frac}} \bigg ) \\ &\qquad +\sqrt \sum _ \in \mathcal (\mathcal )} \alpha (\vec ) \left( H\bigg ( \frac} \bigg ) - 2 H\bigg ( \frac_-)}} \bigg ) + H\bigg ( \frac_\pm )}} \bigg ) \right) \end \end$$

(4.18)

for every \(t > 0\); the second identity in (4.18) holds because for any path \(\vec \in \mathcal (\mathcal )\) one has and , and the map is a bijection on \(\mathcal (\mathcal )\), hence

Proof of Proposition 4.8

(i) follows immediately from (4.10).

(ii) Using Leibniz integral rule once again, for \(\alpha , \beta , \ell \in \mathbb _+\), we deduce that

$$\begin \begin&\int _0^\alpha \int _0^\beta \textrm^} \,\textrmy \,\textrmx \\ &\quad \qquad = \sqrt t\Bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \\&+ H\bigg (\frac} \bigg ) \Bigg ) \end \quad }\, t>0. \end$$

(4.19)

Indeed, differentiating the left-hand side of (4.19) with respect to \(\beta \), we observe at first that

$$\begin \frac}\beta } \int _0^\alpha \int _0^\beta \textrm^} \,\textrmy \,\textrmx&= \int _0^\alpha \textrm^} \,\textrmx = 2 \sqrt \int _}}^}} \textrm^ \,\textrmx \\ &= \sqrt \bigg ( \textrm\bigg (\frac}\bigg ) - \textrm\bigg ( \frac} \bigg ) \bigg ) \end$$

which yields, since \(H'(x) = -\textrm(x)\) for all \(x \in \mathbb \),

$$\begin \int _0^\alpha \int _0^\beta \textrm^} \,\textrmy \,\textrmx&= \sqrt\bigg (\int _0^\beta \textrm\bigg (\frac} \bigg ) \,\textrms - \int _0^\beta \textrm\bigg (\frac} \bigg ) \,\textrms \bigg ) \\ &= \sqrtt \Bigg ( \int _}}^}} \textrm(s) \,\textrms - \int _}}^}} \textrm(s) \,\textrms \Bigg ) \\ &= \sqrtt \Bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \\&\quad + H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \Bigg ) \end$$

implying (4.19) and, according to Lemma 4.7, the expansion in (4.17).

Now as \(\ell (\vec p_-) = \ell (\vec p_+) = 0\), and \(\alpha (\vec ) = 1\) whenever \(\# \vec = 1\), it follows that

$$\begin&\sum __(\mathcal )} \alpha (\vec ) \left( H\bigg ( \frac} \bigg ) - H\bigg ( \frac} \bigg ) - H\bigg ( \frac} \bigg ) + H\bigg ( \frac} \bigg ) \right) \\ &\qquad \quad = 2\sum __(\mathcal )} \left( H\bigg ( \frac} \bigg ) - H( 0 ) \right) = -\frac} \# \textsf+ 4 \sum _\in \textsf} H \bigg (\frac}} \bigg ) \end$$

for all \(t>0\), as \(\# \mathcal _1(\mathcal ) = 2\# \textsf\) and \(H(0) = \frac}\) (see (4.9)), yielding the first identity in (4.18) due to (4.17). \(\square \)

All addends appearing in (4.18) are convergent for fixed \(t>0\), for \(\mathcal \) and \(_}}\subset \textsf\), and we can thus introduce quantities

$$\begin \widetilde_} := \sum _ \in \mathcal (\mathcal )} \alpha (\vec ) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac_-)}} \bigg ) \bigg ), \end$$

(4.20)

and

$$\begin \widetilde_}} := \sum _(\mathcal )} \alpha (\vec p) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \bigg ), \end$$

(4.21)

for \(t>0\) and study them separately, using Lemma 3.6 and Lemma 3.7. More precisely, we compare the scattering coefficient of a fixed directed path \(\vec \) to the sum of corresponding scattering coefficients of its pre-extended directed paths in \(\langle \vec p \rangle _-\). According to Lemma 3.7, this sum equals either \(\alpha (\vec )\) or \(-\alpha (\vec )\), depending on whether \(\vec \) starts at \(_}}\) or \(_}}\). It turns out that directed paths starting at \(_}}\) yield a vanishing contribution, whereas directed paths starting at \(_}}\) yield an additional factor of 2, as stated in the following lemma.

Lemma 4.9

The expressions in (4.20) and (4.21) can be rewritten as

$$\begin \widetilde_} = 2 \sum _ \in __}} \rightarrow }(\mathcal )}} \alpha (\vec ) H\bigg ( \frac)}} \bigg ) - \frac}} \end$$

(4.22)

and

$$\begin \widetilde_}} = 2 \sum ___}} \rightarrow }(\mathcal )}} \alpha (\vec p) H\bigg ( \frac_+)}} \bigg ) - 2\sum _\in \textsf} H\bigg ( \frac}} \bigg ), \end$$

(4.23)

respectively, for all \(t>0\).

Note that, again due to the symmetry , one can replace the set \(__}} \rightarrow }(\mathcal )}\) in (4.22) and (4.23) by \(__}}}(\mathcal )}\) and vice versa.

Proof

We first determine the identity in (4.22): first, as \(\mathcal (\mathcal ) = __}}}(\mathcal )} \cup __}}}(\mathcal )}\), we can write

$$\begin \begin \widetilde_}&= \sum _ \in __}}}(\mathcal )}} \alpha (\vec ) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \bigg ) \\ &\quad \qquad + \sum _ \in __}}}(\mathcal )}} \alpha (\vec ) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \bigg ) \\ &=: \widetilde_^_}}} + \widetilde_^_}}}, \end \qquad }\, t>0. \end$$

(4.24)

According to Lemma 3.6, for \(\textsf\in \_}}, _}}\}\) one has that

$$\begin \begin \sum _ \in _}(\mathcal )}} \alpha (\vec )H\bigg ( \frac_-)}} \bigg )&= \sum _ \in _}(\mathcal )}} \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) H \bigg ( \frac_-)}} \bigg ) \\&\quad + \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} \alpha (\vec ) H \bigg ( \frac_-)}} \bigg ) \\ &= \sum _ \in _}(\mathcal )}} \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) H \bigg ( \frac)}} \bigg ) + \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \end \end$$

(4.25)

for all \(t>0\), because \(\vec _- = \vec \) for all \(\vec \in \langle \vec \rangle _-\) by definition, and \(\ell (\vec _-)=0\) as well as \(\alpha (\vec ) =1\) for all \(\vec \in \mathcal _1(\mathcal )\). Thus, due to the convergence in Proposition 4.8 for any \(t>0\), we can decompose \(\widetilde_^\textsf}\), for \(\textsf\in \_}},_}}\}\) as

$$\begin \begin \widetilde_^\textsf}&= \sum _ \in _}(\mathcal )}} \alpha (\vec ) H\bigg (\frac} \bigg ) - \sum _ \in _}(\mathcal )}} \alpha (\vec ) H\bigg (\frac} \bigg ) \\ &= \sum _ \in _}(\mathcal )}} \Bigg ( \alpha (\vec ) - \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) \Bigg ) H\bigg (\frac)}} \bigg ) - \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac}, \end \end$$

(4.26)

for every \(t>0\). Now, by Lemma 3.7, we have that

$$\begin \alpha (\vec ) - \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) = 2\alpha (\vec ), & }\, \textsf= _}}, \\ 0, & }\, \textsf= _}}, \end\right. } \qquad }\, \vec \in _ \rightarrow }(\mathcal )}, \end$$

(4.27)

therefore, we obtain for \(\widetilde_^_}}}\) and \(\widetilde_^_}}}\) that

$$\begin \widetilde_^_}}}&= 2\sum _ \in __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )}} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \\ &= 2\sum _ \in \mathcal __}}}(\mathcal )} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \end$$

and

$$\begin \widetilde_^_}}}&= 2\sum _ \in __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )}} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \\ &= 2\sum _ \in \mathcal __}},_}}}(\mathcal )} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \end$$

for all \(t>0\), yielding (4.22) by (4.24), since

$$\begin \begin \mathcal __}}}(\mathcal ) \sqcup \mathcal __}},_}}}(\mathcal )&= \Big ( __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )} \Big ) \sqcup \Big ( __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )} \Big ) \\ &= \underbrace__}}}(\mathcal )} \sqcup __}}}(\mathcal )} \big )}_(\mathcal )} \cap \, __}} \rightarrow }(\mathcal )} = __}} \rightarrow }(\mathcal )}; \end \end$$

(4.28)

likewise \(\big (__}}}(\mathcal )} \cap \mathcal _1(\mathcal )\big ) \sqcup \big (__}}}(\mathcal )} \cap \mathcal _1(\mathcal )\big ) = \mathcal _1(\mathcal )\), and consequently,

$$ \sum _ \in \mathcal _1(\mathcal )} \frac} = \sum _} \in \textsf} \frac} = \frac}} = \frac}}. $$

We determine (4.23) in a similar manner: again, we decompose

$$\begin \begin \widetilde_}}&= \sum _ \in __}}}(\mathcal )}} \alpha (\vec ) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \bigg ) \\ &\quad \qquad + \sum _ \in __}}}(\mathcal )}} \alpha (\vec ) \bigg ( H\bigg (\frac} \bigg ) - H\bigg (\frac} \bigg ) \bigg ) \\ &=: \widetilde_^_}}}} + \widetilde_^_}}}}, \end \qquad }\, t>0. \end$$

(4.29)

and, using again Lemma 3.6, write

$$\begin \begin \sum _ \in _}(\mathcal )}} \alpha (\vec )H\bigg ( \frac_\pm )}} \bigg )&= \sum _ \in _}(\mathcal )}} \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) H \bigg ( \frac_\pm )}} \bigg ) \\&+ \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} \alpha (\vec ) H \bigg ( \frac_\pm )}} \bigg ) \\ &= \sum _ \in _}(\mathcal )}} \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) H \bigg ( \frac_+)}} \bigg ) + \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg ( \frac)}} \bigg ) \end \end$$

(4.30)

for \(\textsf\in \_}}, _}}\}\) and all \(t>0\), as \(\vec _\pm = (\vec _-)_+ = \vec _+\) for any \(\vec \in \langle \vec \rangle _-\), and for all \(\vec \in \mathcal _1(\mathcal )\), thus and by definition. Hence, like in (4.26), this implies

$$\begin \begin \widetilde_^\textsf}}&= \sum _ \in _}(\mathcal )}} \alpha (\vec ) H\bigg (\frac} \bigg ) - \sum _ \in _}(\mathcal )}} \alpha (\vec ) H\bigg (\frac} \bigg ) \\ &= \sum _ \in _}(\mathcal )}} \Bigg ( \alpha (\vec ) - \sum _ \in \langle \vec \rangle _-} \alpha (\vec ) \Bigg ) H\bigg (\frac_+)}} \bigg ) - \sum _ \in _}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg ( \frac)}} \bigg ), \end \end$$

(4.31)

for \(\textsf\in \_}},_}}\}\) and all \(t>0\) and again by (4.27) this leads to the expressions

$$\begin \widetilde_^_}}}}&= 2\sum _ \in __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )}} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg (\frac)}} \bigg ) \\ &= 2\sum _ \in \mathcal __}}}(\mathcal )} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg (\frac)}} \bigg ) \end$$

and

$$\begin \widetilde_^_}}}}&= 2\sum _ \in __}}}(\mathcal )} \cap __}} \rightarrow }(\mathcal )}} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg (\frac)}} \bigg ) \\ &= 2\sum _ \in \mathcal __}},_}}}(\mathcal )} \alpha (\vec )H\bigg (\frac} \bigg ) - \sum _ \in __}}}(\mathcal )} \cap \mathcal _1(\mathcal )} H\bigg (\frac)}} \bigg ) \end$$

for all \(t>0\). These expressions for \(\widetilde_^_}}}}\) and \(\widetilde_^_}}}}\), respectively, then imply (4.23) by (4.29), using the same decompositions as in (4.28) and afterward, noting that

$$ \sum _ \in \mathcal _1(\mathcal )} H \bigg (\frac)}} \bigg ) = \sum _} \in \textsf} H \bigg (\frac}} \bigg ) = 2\sum _\in \textsf} H \bigg (\frac}} \bigg ); $$

this finishes the proof. \(\square \)

Following the idea described right before Lemma 4.9, applied on the term

$$ \sum _ \in __}} \rightarrow }(\mathcal )}} \alpha (\vec p)\Bigg ( H\bigg ( \frac)}} \bigg ) - H\bigg ( \frac_+)}} \bigg ) \Bigg ), $$

and arguing in terms of post-extended rather than pre-extended paths, we are now in the position to present the proof of our main result, Theorem 4.1.

Proof of Theorem 4.1

Let us recall that we are going to show that

$$\begin \mathcal _t&= \vert \mathcal \vert - \frac}} \# _}}+ 4\sqrt \sum _ \in \mathcal __}}(\mathcal )} \alpha (\vec p) \, H\bigg ( \frac} \bigg ). \end$$

(4.32)

Plugging (4.20) and (4.21) in (4.18), and then applying Lemma 4.9, we obtain

$$\begin \begin \mathcal _t&= \vert \mathcal \vert +\frac}} \# \textsf- 2\sqrt\sum _\in \textsf} H \bigg ( \frac}} \bigg ) + \sqrt \Big (\widetilde_} - \widetilde_}} \Big ) \\ &= \vert \mathcal \vert + 2\sqrt \sum _ \in __}} \rightarrow }(\mathcal )}} \alpha (\vec p)\Bigg ( H\bigg ( \frac)}} \bigg ) - H\bigg ( \frac_+)}} \bigg ) \Bigg ), \end \end$$

(4.33)

for \(t>0\). Now, by Lemma 3.6, we can write

$$\begin \sum _ \in __}} \rightarrow }(\mathcal )}} \alpha (\vec p)H\bigg ( \frac_+)}} \bigg )&= \sum _ \in __}} \rightarrow }(\mathcal )}} \sum _ \in \langle \vec \rangle _+} \alpha (\vec ) H\bigg ( \frac_+)}} \bigg ) \\&+ \sum _ \in __}} \rightarrow }(\mathcal )} \cap \mathcal _1(\mathcal )} \alpha (\vec ) H\bigg ( \frac_+)}} \bigg ) \\ &= \sum _ \in __}} \rightarrow }(\mathcal )}} \sum _ \in \langle \vec \rangle _+} \alpha (\vec ) H\bigg ( \frac)}} \bigg ) + \sum _ \in __}} \rightarrow }(\mathcal )} \cap \mathcal _1(\mathcal )} \frac}, \end$$

for every \(t>0\), since \(\vec _+ = \vec \) for any \(\vec \in \langle \vec \rangle _+\), and \(\ell (\vec _+) = 0\) as well as \(\alpha (\vec )=1\) for every \(\vec \in \mathcal _1(\mathcal )\). Moreover, by Lemma 3.7

$$\begin \alpha (\vec ) - \sum _ \in \langle \vec \rangle _+} \alpha (\vec ) = 2\alpha (\vec ), & }\, \vec \in __}}}(\mathcal )}, \\ 0, & }\, \vec \in __}}}(\mathcal )}, \end\right. } \qquad }\, \vec \in __}} \rightarrow }(\mathcal )}. \end$$

(4.34)

Using Proposition 4.8.(i) and, again, \(H(0)=\frac}\), this eventually leads to

$$\begin \begin&\sum _ \in __}} \rightarrow }(\mathcal )}} \alpha (\vec p) \Bigg ( H\bigg ( \frac)}} \bigg ) - H\bigg ( \frac_+)}} \bigg ) \Bigg ) \\ \\ &\qquad \quad = \sum _ \in __}} \rightarrow }(\mathcal )}} \Bigg ( \alpha (\vec ) - \sum _ \in \langle \vec \rangle _+} \alpha (\vec ) \Bigg ) H\bigg ( \frac)}}\bigg ) -\sum _ \in __}} \rightarrow }(\mathcal )} \cap \mathcal _1(\mathcal )} \frac} \\&\qquad \quad = 2\sum _ \in __}} \rightarrow }(\mathcal )} \cap __}}}(\mathcal )}} \alpha (\vec )H\bigg ( \frac)}}\bigg ) - \sum _} \in \textsf, \, \partial ^-(\vec }) \in _}}} \frac} \\&\qquad \quad = 2\sum _ \in \mathcal __}}(\mathcal )} \alpha (\vec )H\bigg ( \frac)}}\bigg ) - \frac_}}}}. \end \end$$

(4.35)

Now plugging (4.35) into (4.33) finally yields (4.32), and thus the claimed formula in (4.1). \(\square \)

Example 4.10

(Lasso graph). Let us examine (4.1) where \(\mathcal \) is a lasso graph (see Fig. 12) with a single Dirichlet vertex \(_}}= \_\textrm \}\) connected to an edge \(\textsf_1\) of length \(\ell _1 >0\) with a standard vertex \(_}}\) at the other end which is connected to a loop \(\textsf_2\) of length \(\ell _2 > 0\):

Fig. 12

A lasso graph with a Dirichlet condition at \(_}}\)

Given \(\vec p \in \mathcal (\_}}\})\), i.e., a directed path starting and ending at \(_}}\), we can count the number

\(R_(\textsf_\textrm)\) of times \(\vec \) bounces back at \(_}}\);

\(R_^(_}})\) of times \(\vec \) traverses \(\textsf_1\), hits \(_}}\), and bounces back into \(\textsf_1\);

\(R_^(_}})\) of times \(\vec \) traverses \(\textsf_2\), hits \(_}}\), and bounces back into \(\textsf_2\);

\(T_^(_}})\) of times \(\vec \) traverses \(\textsf_1\) and is transferred into \(\textsf_2\) through \(_}}\);

\(T_^(_}})\) of times \(\vec \) traverses \(\textsf_2\) and is further transferred into \(\textsf_2\) through \(_}}\),

respectively. This leads to a length of

$$ \ell (\vec p) = \big (R_}(\textsf_\textrm) + R_}^(_}}) +T_^(_}})+ 1 \big )\ell _1 + \big ( T_}^(_}}) + R_}^(_}}) + T_}^(_}}) \big ) \ell _2 $$

and a scattering coefficient

$$\begin \alpha (\vec p)&= (-1)^(\textsf_\textrm)}\bigg (-\frac\bigg )^^(_}}) + R_^(_}})} \bigg (\frac\bigg )^^(_}}) + T_^(_}})}. \end$$

Note that whenever \(\vec p\) is transferred into \(\textsf_2\) after traversing \(\textsf_1\), it must later arrive at \(\textsf_1\) again (via \(\textsf_2\)) as \(\vec p\) has to finish at \(_}}\). Hence, each time \(\vec p\) arrives along \(\textsf_1\) and is further transferred into \(\textsf_2\), this contributes a factor of \(\frac\) to \(\alpha (\vec p)\) twice.

Letting \(L_1(\vec ):= R_}(\textsf_\textrm) + R_}^(_}}) +T_^(_}}) +1\) and \(L_2(\vec ):= T_}^(_}}) + R_}^(_}}) + T_}^(_}})\) the heat content formula (4.1)—which is based on path enumeration—at first yields

$$\begin \mathcal _t(\mathcal };\_\textrm \})&= \ell _1 + \ell _2 - \frac}} \\&\quad + 4\sqrt \sum _(\_\textrm \})} (-1)^(\textsf_\textrm)}\bigg (-\frac\bigg )^^(_}}) + R_^(_}})} \\&\quad \bigg (\frac\bigg )^^(_}}) + T_^(_}})} H\bigg ( \frac)\ell _1 + L_2(\vec ) \ell _2}} \bigg ). \end$$

Moreover, using some combinatorics, it is possible to rewrite the above formula as

$$\begin \begin \mathcal _t(\mathcal };\_\textrm \})&= \ell _1 + \ell _2 - \frac}} \\&+ 4\sqrt \sum _^\infty \sum _^ (-1)^\bigg (-\frac\bigg )^ \bigg (\frac\bigg )^ \left( 2m+1\\ n\end}\right) \left( 2m-n+1\\ m-n+1\end}\right) H_(t), \end \end$$

(4.36)

where the function \(H_\), \(m \in \mathbb _0\), \(n =0,\dots ,m+1\), given by

$$\begin H_(t) := H\Big ( \frac} \Big ), & }\, m+1=n, \\ \sum \limits _^ 2^ \big (-\frac\big )^ \big (\frac \big )^ H\Big ( \frac} \