2.1 A fixed-point construction

Let B be a sub-graph of a simple graph X. We define the set $\operatorname {Con}_X(B)$ to be the set of connected components in $X \setminus B$ . Associated with B, we define the graph $X_B$ with vertex set $V(B)$ and an edge $\{x, y\}$ if there is a component in $\operatorname {Con}_X(B)$ with both x and y in its boundary in X. Roughly, they are both incident to a common component created by deleting B from X. We give a simple example below.

Definition 2 (Self-similar with respect to a sub-graph, and a map)

A connected infinite graph X is said to be self-similar with respect to sub-graph B and a map

$$\begin{align*}\psi: V(X) \to V(X_B)\end{align*}$$

if the following hold:

1. (1) no two vertices in B are adjacent in X;

2. (2) the intersection of the boundaries of two different components in $\operatorname {Con}_X(B)$ contains at most one vertex;

3. (3) $\psi (X)=X_B$ is isomorphic to X.

The components of $\operatorname {Con}_X(B)$ are called cells. Consider a map $\psi $ that replaces a cell $\hat C$ in X with a complete graph on the vertices of $\theta C$ . If $\psi (X)$ is isomorphic to X, then we have self-similarity.

Example 1 Consider the infinite finite path graph X with every second vertex colored blue.

Let $B\subset X$ be the set of blue vertices, then $\operatorname {Con}_X(B)$ is the set of (isolated) white vertices. These are the cells; remark that the cells are all isomorphic. The closure of a cell includes the two adjacent blue vertices. Each looks like:

Let $\psi $ be the map on X that replaces every instance of $\hat C$ in X with a single edge between the blue vertices. The image of $\psi $ is again an infinite finite path graph, and is hence in bijection with X. Thus, we can say that X is self-similar with respect to the set of blue vertices and $\psi $ .

We see that the finite path graph in the example is symmetrically self-similar as the additional conditions are met.

Under these conditions, an origin cell is a cell C such that $\psi (\theta C) \subset C$ . A vertex fixed point of $\psi $ is called an origin vertex. If X is self-similar with respect to B and $\psi ,$ then by Krön, [Reference Krön9, Theorem 1] X has precisely one origin cell and no origin vertex or exactly one origin vertex, which we shall denote $\mathbf {o}$ when it is the case.

As the cells are isomorphic, we can talk about the cell of a symmetrically self-similar graph. Denote this cell by C and its boundary in X by $\theta C$ . The boundary vertices $\theta C$ are called the extremal vertices. If there are $\theta $ of them, then $\theta $ is the branching number of X. If we write $\hat C$ as the cell associated with the graph, we are referring to the closure in X.

It can be shown that the edges of $\hat C$ can be partitioned into $\mu $ complete graphs on $\theta $ vertices, and that $\psi $ maps instances of $\hat C$ to complete graphs on the $\theta $ extremal vertices of $\hat C$ (i.e., $\psi (\hat C)$ is isomorphic to $K_\theta $ ). By simple accounting, we have $\mu = \frac {2|E|}{\theta (\theta -1)}$ and so $\mu $ corresponds to the usual mass scaling factor of self-similar sets. Figure 2 has three examples of cell graphs (indicating the extremal vertices in blue), and showing the values of $\theta $ and $\mu $ . If a self-similar graph is bipartite, its cell graph is bipartite and $\theta =2$ .

2.2 A blowup construction

Krön and Teufl [Reference Krön and Teufl10, Theorem 1] describe the reverse of this construction as an iterative process whose limit is a symmetrically self-similar graph. The initial graph is $\hat C$ , and at each step, the next graph is generated using an inverse action to $\psi $ : the $\mu $ copies of $K_\theta $ in $\hat C$ are “blown up” and are replaced by a copy of the $\hat C$ . The sequence of substitutions converges to a unique graph X, which is fixed under $\psi $ . The first few iteration that builds the Sierpiński graph is pictured in Figure 4.

Figure 4

$C_1=\hat C$ and $C_2$ . From $C_i$ to $C_{i+1}$ , the shaded triangles are each replaced by a copy of $\hat C$ . The limit graph in this case is the Sierpiński graph.

Remark that the graph with a finite number of copies of this graph centered at $\mathbf {o}$ is also a fixed point of $\psi $ . That is, given $\hat C$ , we can generate a countable family of symmetrically self-similar graphs. However, remark that as copies are added, the outward probability of the origin is reduced by the same factor, and hence the Green’s functions will all be the same.

3.1 Finite graphs

The study of Green’s functions of finite graphs is a classic topic, going back to at least the 1950s. We will recall some results as presented in [Reference Flajolet and Sedgewick7], as they are very helpful for our purpose. Very roughly, in the case of a finite graph, we can express Green’s functions as a quotient of two determinants of the probability matrix. Thus, in this case, the Green’s function will be a rational function.

Proposition 4 (Application of Proposition V.6 of Flajolet and Sedgewick)

Let W be a digraph on the vertex set $\{v_1, \dots , v_k\}$ and let T be the $k\times k$ matrix of edge weights $T[i,j]=\operatorname {weight}(v_i,v_j)$ . The weight of a path is the product of the weights of the edges in the path. The series $F_{i,j}(z)$ defined so the coefficient of $z^n$ is the sum of the weights of all paths from $v_i$ to $v_j$ of length n in W, is the entry $[i,j]$ of the matrix $(I-zT)^{-1}$ . Otherwise stated,

(3) $$ \begin{align} F_{i,j}(z)=\left( (1-zT)^{-1}\right)=(-1)^{i+j}\frac{\Delta^{j,i}(z)}{\Delta(z)}. \end{align} $$

Here, $\Delta (z)=\det (I-zT)$ and $\Delta ^{j,i}(z)$ is the determinant of the minor of index $j,i$ of $I-zT$ .

Thus, in the finite case, the Green’s function is a rational function. We can deduce further information about a singular expansion near its dominant singularity. Recall that matrices constructed this way are said to be irreducible if the graph system is strongly connected, that is, there is a path between any two vertices.

Lemma 5 (Lemma V.1 of [Reference Flajolet and Sedgewick7] (Iteration of irreducible matrices))

Let the nonnegative matrix T be irreducible and aperiodic, with $\lambda _1$ its dominant eigenvalue. Then, the residue matrix such that

$$\begin{align*}(I-zT )^{-1} = \frac{\Phi}{1-z\lambda_1} +O(1) \quad(z \rightarrow \lambda^{-1})\end{align*}$$

has entries given by $\phi _{i,j} = \frac {r_i\ell _j}{\langle r, \ell \rangle }$ , where r and $\ell $ are, respectively, right and left eigenvectors of T corresponding to the eigenvalue $\lambda _1$ .

Theorem 6 (Theorem V.7 of [Reference Flajolet and Sedgewick7] (Asymptotics of paths in finite graphs))

Consider the matrix

$$\begin{align*}F(z)=\left(I-zT \right)^{-1},\end{align*}$$

where T is a scalar nonnegative matrix, in particular, the adjacency matrix of a graph equipped with positive weights. Assume that T is irreducible. Then, all entries $F_{i,j}(z)$ of $F(z)$ have the same radius of convergence $\rho $ , which can be defined in two equivalent ways:

1. (1) as $\rho =\lambda _1^{-1}$ , where $\lambda _1$ is the largest positive eigenvalue of T;

2. (2) as the smallest positive root of the determinantal equation: $\det (I-zT )=0$ .

Furthermore, the point $\rho =\lambda _1^{-1}$ is a simple pole of each $F_{i,j}(z)$ . If T is irreducible and aperiodic, then $\rho =\lambda _1^{-1} $ is the unique dominant singularity of each $F_{i,j}(z)$ , and

$$\begin{align*}[z^n]F_{i,j}(z)=\phi_{i,j} \lambda_1^n +O(\Lambda^n), 0 \leq\Lambda<\lambda_1,\end{align*}$$

for computable constants $\phi _{i,j}>0$ .

Given a finite graph with vertex set $\{v_1, \dots v_k\}$ if we set T to be P, the matrix of transition probabilities as defined in Equation (1), then $(T^n)_{ij}=p^{(n)}(v_i,v_j)$ , and $F_{ij}(z)=G(v_i, v_j|z)$ . From this result, we can understand why all Green’s functions of a fixed finite graph have the same radius of convergence: they are all the multiplicative inverse of the dominant eigenvalue of T.

Since our cells are connected undirected graphs, T is irreducible, and it is only periodic when the cell is bipartite. We have a good understanding of this case, and principally, we can show (by considering $T^2$ , which will not be periodic) that in the value we are interested in:

$$\begin{align*}[z^n]F_{1,1}(z)=\begin{cases} \phi_{1,1} \lambda_1^n +O(\Lambda^n), 0 \leq\Lambda<\lambda_1, & n\text{ even}\\ 0, & \text{otherwise.} \end{cases}\end{align*}$$

Recall the following two facts from classic matrix theory. If T is a stochastic matrix, then 1 is a dominant eigenvalue of T. If T is a sub-stochastic matrix, that is, there is at least one rowsum strictly less than 1, then the dominant eigenvalue is strictly less than 1, and hence the dominant singularity of any $F_{i,j}(z)$ will be strictly greater than 1 since it is the multiplicative inverse of the dominant singularity.

Example 2 (Diamond)

Consider the labeled graph and its transition matrix given in Figure 5.

Figure 5

A graph and its transition matrix.

This matrix has two dominant eigenvalues 1 and $-$ 1, and we can see that

$$\begin{align*}G_{\hat C}(v_1, v_1|z)=F_{1,1}(z)=\frac{z^{4}-9 z^{2}+9}{3 \left(z^{2}-3\right) \left(z^{2}-1\right)}. \end{align*}$$

The series starts $G(v_1, v_1|z)= 1+\frac {1}{3} z^{2}+\frac {2}{9} z^{4}+\frac {5}{27} z^{6}+\frac {14}{81} z^{8}+\mathrm {O}\! \left (z^{10}\right )$ . For example, we see that $p^{(8)}(v_1, v_1)=\frac {14}{81}$ . Remark that the periodicity is evident as $G_{\hat C}(v_1, v_1| z)$ is a function of $z^2$ . This is a direct consequence of the bipartiteness of the graph.

In the next section, we apply these theorems to the cells of a symmetrically self-similar graph, setting some edge weights to 0.

3.2 Symmetrically self-similar graphs

Krön and Teufl translate the iterative fractal nature of X into a functional equation for the Green’s function in one of the main results of [Reference Krön and Teufl10], exploiting the iterative nature of the blow up construction. Define d to be the transition function: the generating function of the probabilities that the simple random walk on $\hat C$ starting at $\mathbf {o}$ in $\theta C$ hits a vertex in $\theta C\setminus \{\mathbf {o}\}$ for the first time after exactly n steps. It may return to the origin before hitting this other cell.

Next, define f to be the return function: the generating function of $f_n$ , the probability that the random walk on $\hat C$ starting at $\mathbf {o}$ returns to $\mathbf {o}$ after n stages without hitting a vertex in $\theta C\setminus \{\mathbf {o}\}$ . Since the start is considered the first visit, $f_0=1$ , and thus $f(0)=1$ . We also use $r(z)$ , the probability function of first return which satisfies $f(z)=\frac {1}{1-r(z)}$ .

Because $\hat C$ is finite, f and d are expressed in terms of Green’s functions of a finite graph. They are well understood: they are rational functions that can be computed as the determinant of a specific matrix. We first give the functional equation, and then consider singular expansions of G, $d,$ and f.

Given the closed cell $\hat C$ , and let $P_{\hat C}$ be the probability transition matrix associated with it. Set $P_f$ to be the matrix $P_{\hat C}$ with the modification $P_{f}[i, j]:=0$ for $1<j\leq \theta $ , $i>\theta $ . Then, $f(z)=\left [(I-zP_f)\right ]^{-1}[1,1]$ . Set $P_d$ to be the matrix $P_{\hat C}$ with the modification $P_{d}[i,j]:=0$ for $1<i\leq \theta $ , $j>\theta $ . Then, $d(z)=\sum _{j=2}^\theta \left [(1-zP_d)\right ]^{-1}[1,j]$ . Thus, both f and d can be determined by straightforward matrix inverse computations (possibly followed by some sums).

Lemma 7 (Krön and Teufl [Reference Krön and Teufl10, Lemma 3])

Let $G(z)$ be the Green’s function for a symmetrically self-similar graph with origin $\mathbf {o}$ . Then, the following equation holds for all $z \in U(0,1)$ :

(4) $$ \begin{align} G(z) = f(z)G(d(z)). \end{align} $$

This functional equation was first established and exploited in the case of the Sierpiński graph by Grabner and Woess [Reference Grabner and Woess8] before being generalized for all self-similar graphs by Krön [Reference Krön9].

Example 4 (Diamond example continued)

We can define f and d for the diamond example using the following matrices of graph weights:

(5) $$ \begin{align} P_f=\left[\begin{array}{cccccc} 0& 0& 1 & 0 & 0& 0\\ 0 & 0 & 0&1 & 0& 0 \\ \frac13 & 0 & 0 &0&\frac13 &\frac13\\ 0& 0&0&0& \frac13&\frac13\\ 0&0&\frac12&\frac12&0&0\\ 0&0&\frac12& \frac12&0&0 \end{array}\right] \quad P_d=\left[\begin{array}{cccccc} 0& 0& 1 & 0 & 0& 0\\ 0 & 0 & 0&0 & 0& 0 \\ \frac13 & 0 & 0 &0&\frac13 &\frac13\\ 0& \frac13&0&0& \frac13&\frac13\\ 0&0&\frac12&\frac12&0&0\\ 0&0&\frac12& \frac12&0&0 \end{array} \right]. \end{align} $$

It follows that

$$ \begin{align*} f(z) &= (1-zP_f)^{-1}[1,1] =-\frac{3 \left(2 z^{2}-3\right)}{z^{4}-9 z^{2}+9}=1+\frac{1}{3} z^{2}+\frac{2}{9} z^{4}+\mathrm{O}\! \left(z^{10}\right)\\ \text{and}\quad d(z)&= (1-zP_d)^{-1}[1,2]=\frac{z^{4}}{z^{4}-9 z^{2}+9} =\frac{1}{9} z^{4}+\frac{1}{9} z^{6}+\frac{8}{81} z^{8}+\mathrm{O}\! \left(z^{10}\right). \end{align*} $$

The dominant eigenvalues of both $P_f$ and $P_d$ are $\frac {1+\sqrt {5}}{2\sqrt {3}}$ and hence $\rho _f=\rho _d = \frac {2\sqrt {3}}{1 + \sqrt {5}} \approx 1.070$ . Using these, we can determine the initial series expansion of the Green’s function,

(6) $$ \begin{align} G(\mathbf{o}, \mathbf{o}|z) \equiv G(z)&= f(z)\,f(d(z))\,f(d(d(z)))\dots \end{align} $$

(7) $$ \begin{align} &=1+\frac{1}{3} z^{2}+\frac{2}{9} z^{4}+\frac{5}{27} z^{6}+O(z^7).\end{align} $$

We inspect the components of this equation more closely. The Green’s function $G(u,v|z)$ of a finite graph with transition probability matrix P is the corresponding $[u,v]$ entry of $(I-zP)^{-1}$ . Analytically, we know a fair amount about G in this case. Remark that the result does not apply directly to bipartite graphs, as P fails the condition of aperiodicity, but there are workarounds using $P^2$ , for example. A key result is the singular expansion of f (and d) is of the form $\kappa (1-z/\rho )^{-1} + O(1)$ , as $z\rightarrow \rho $ for a real, positive $\kappa $ . Furthermore, as $\hat C$ is finite, d and f are rational.

A graph with bounded geometry here refers to a graph where the number of neighbors each vertex has is finite and has a uniform upper bound. In the case of symmetrically self-similar graphs with bounded geometries, Krön and Teufl exploit their functional equation to determine the singular expansion of $G(\mathbf {o},\mathbf {o}|z)$ near $z=1$ . Here, $\tau =d'(1)$ and $\alpha =f(1)$ . From their combinatorial definition, note that both quantities are greater than 1.

Theorem 8 (Krön and Teufl [Reference Krön and Teufl10, Theorem 5])

Let X be a symmetrically self-similar graph with bounded geometry and origin vertex $\mathbf {o}$ . Then, there exists a 1-periodic, holomorphic function $\omega $ on some horizontal strip around the real axis such that the Green’s function G has the local singular expansion

(8) $$ \begin{align} \forall |z|<1, \quad G(z) = (z-1)^{\eta} \left( \omega \left( \frac{1-z}{\tau} \right) + o(z-1) \right) \end{align} $$

with $\eta = \frac {\log \mu }{\log \tau }-1$ .

Recall that $\mu $ is the number of cliques that form $\hat C$ . When $\theta =2,$ this is the number of edges in $\hat C$ . Moreover, even though it is not directly stated, it follows from a quick analysis of the asymptotics of G at $1$ that $\eta = -\frac {\log \alpha }{\log \tau }$ . Because $\alpha $ and $\tau $ are both greater than 1, $\eta $ is a negative number.

Example 5 (Diamond example continued)

Since we have exact expressions for f and $d,$ we compute $\eta $ :

$$\begin{align*}\eta = \frac{\log\mu}{\log\tau}-1=\frac{\log 6}{\log 18}-1=-\frac{\log 3}{\log18}=\frac{f(1)}{d'(1)}\approx -0.3800. \end{align*}$$

The domain of analyticity of G is governed by the dynamic behavior of f and d, according to Krön [Reference Krön9] and Krön and Teufl [Reference Krön and Teufl10]. If $\omega $ is not constant, a singularity analysis shows that G is not algebraic. Krön and Teufl conjecture that: if $\omega $ is constant, then $\hat C$ is a path graph.

Using a result of Di Vizio, Fernandes, and Mishna, which appeared recently in a preprint [Reference Di Vizio, Fernandes and Mishna5], we can say more.

Theorem 9 (Vizio, Fernandes, and Mishna [Reference Di Vizio, Fernandes and Mishna5, Special case of Theorem C])

Suppose $y(z)\in \mathbb {C}[[z]]$ is a series that satisfies the following iterative equation with some $a, b\in \mathbb {C}(z)$ :

$$\begin{align*}y(z)=a(z)y(b(z)).\end{align*}$$

If $b(z)\in \mathbb {C}(z)$ satisfies the following conditions: $b(0) = 0$ ; $b'(0) \in \{ 0,1, \text {roots of unity}\}$ ; and no iteration of $b(z)$ is equal to the identity, then either there is some $ N \in \mathbb {N}^*$ , such that $y(z)^N$ is rational, or y is differentially transcendental over $\mathbb {C}(z)$ .

Putting these results together, we have our workhorse result.

Corollary 10 Let $G(z)$ be the Green’s function of a symmetrically self-similar graph with origin $\mathbf {o}$ . Then, $G(z)$ is either algebraic or differentially transcendental over $\mathbb {C}(z)$ . If G is algebraic, then there exists a minimal N such that $G^N=P/Q$ with $P, Q\in \mathbb {C}[z]$ , P and Q co-prime and Q monic. In this case, the equality

(9) $$ \begin{align} f(z)^N\frac{P(d(z))}{Q(d(z))}=\frac{P(z)}{Q(z)} \end{align} $$

is defined and true for all complex z except possibly a finite set of points.

3.3 A few technical results on f and d

The derivative $d'(1)$ is called $\tau $ and is the expected number of steps for a walk on $\hat C$ that starts at $\mathbf {o}$ to hit another external vertex. By definition, external vertices are not adjacent, so $d'(1)>2$ . Since $d_n$ , ( $f_n$ and $r_n$ ) are all probabilities, and hence nonnegative numbers, by Pringsheim’s theorem, the radius of convergence $\rho _d$ (respectively $ \rho _f$ and $\rho _r$ ) of d, (f and r) are singularities. Now, as d is a probability generating function, it follows that $d(1)=1$ and thus $\rho _d>1$ since d converges at 1. As these are Green’s functions, we can deduce key facts about these elements that we use in subsequent proofs. The proofs are a mix of basic facts on Green’s functions and series analysis. The results are demonstrated in Example 4.

4.1 The structure of an algebraic $G(z)$

Krön and Teufl [Reference Krön and Teufl10, p. 13] showed that if X is not bipartite, then $z = 1$ is the only singularity of $G,$ the boundary of the unit disk and if X is bipartite, then the only singularities on the unit disk are $1$ and $-1$ . We start by extending this result in the case that G is algebraic.

Proof If G is algebraic, then $\omega $ in Theorem 8 is constant and by Krön [Reference Krön9, Theorem 7] the singularities of $G(z)$ are real, and contained in the set $(-\infty , -1]\cup [1, \infty )$ . Writing $G^N=P/Q,$ we have that the singularities of G are contained in the roots of P and Q, as these are the poles and zeros of $G^N$ .

Toward a contradiction, assume there is some real $z_0$ , with $z_0>1$ that is either a pole or a zero of $G^N$ . Since $G^N$ has only a finite number of zeros and poles, we can assume that $z_0$ is the smallest one strictly greater than $1$ . Since $\rho _d$ is a pole of $d(z)$ , $d((1,\rho _d)) = (1,\infty )$ , and so there exists a $z' \in (1,\rho _d)$ such that $d(z') = z_0$ . Similarly, f is analytic and defined on the interval $[1, \rho _d)$ . By Lemma 11, $\rho _d=\rho _f$ , so $z'$ is not a pole of f. Since $f(0)=f_0=1$ (by convention, the length of a walk whose first contact with the origin for a walk that starts at the origin is 0) and f is increasing on this interval, so $z'$ is not a zero of f either.

From Equation (9), $\frac {G(z)^N}{f(z)^N} = G(d(z))^N$ , and so

(10) $$ \begin{align} \lim_{z\rightarrow z_0}G(z)^N = \lim_{z\rightarrow z'}G(d(z))^N =\lim_{z\rightarrow z'} \frac{G(z)^N}{f(z)^N}. \end{align} $$

Hence, as $\lim _{z\rightarrow z'} f(z)=f(z')$ is finite and nonzero, if $z_0$ is a zero or pole of $G^N$ greater than 1, then $z'$ is also a zero or pole, but is closer to 1, a contradiction.

This contradiction establishes that there is no real-valued zero or pole greater than 1.

From the singular expansion of G at 1 of Krön and Teufl, we have that 1 is the dominant singularity, and hence there is no other singularity of smaller modulus. Consequently, there is no positive real pole of $G^N$ less than 1.

Since G is defined as a series with nonnegative coefficients, any evaluation of $G(z)$ at a positive real number in its domain of convergence is nonzero, and similarly for $G^N(z)$ . Again, from the singular expansion, we can exclude the $G(1)=0$ case since $\alpha $ and $\tau $ are both greater than 1.

Proof We start with Equation (9) and consider expansions of G, f, and d around $\rho _f=\rho _d$ . Now, $G(z)$ is not singular nor 0 at $\rho _f$ , since $\rho _f>1$ by Lemma 14. Here, $\kappa _f$ and $\kappa _d$ are positive real constants by [Reference Flajolet and Sedgewick7, Theorem 7]. Let $u=(1-z/\rho _f)^{-1}$ . We compute the following limits in $\mathbb {R}$ :

$$ \begin{align*} G(\rho_f)^N &=\lim_{z \rightarrow \rho_f} G(z)^N \\ &=\lim_{z \rightarrow \rho_f} f(z)^N \frac{P(d(z))}{Q(d(z))}\\ &=\lim_{z \rightarrow \rho_f}\left(\frac{\kappa_f^N}{(1-z/\rho_f)^{N}} + O(z-\rho_f)^{-N+1}\right) \frac{P(\kappa_d(1-z/\rho_f)^{-1} + O(1))}{Q(\kappa_d(1-z/\rho_f)^{-1} + O(1))}\\ &=\lim_{u\rightarrow \infty} (\kappa_f^N u^{N} + O(u)^{N-1}) \frac{P(\kappa_d u + O(1))}{Q(\kappa_d u + O(1))}\\ &=\lim_{u\rightarrow \infty} \kappa_f^N u^{N} \frac{P(\kappa_d u)}{Q(\kappa_d u)}. \end{align*} $$

Remark that as z approaches $\rho _f$ , u is arbitrarily large. The degree of Q as a polynomial in z must be larger than N since the value of the last limit is nonzero.

4.2 Results when $\theta =2$

The proof is a little technical and is delayed to the next section. Recall that $\mu $ is the number of edges in $\hat C$ , and $\alpha $ is the expected number of visits to $v_1$ before a visit to $v_2$ . The proof but relies on a straightforward intuition that among all finite cell graphs $\hat C$ where $v_1$ and $v_2$ are a fixed distance apart, the average number of times a random walk starting at $v_1$ hits $v_1$ before $v_2$ (i.e., $f(1)$ ) is minimized when the graph is a line.

Corollary 17 If $\theta =2$ , then $\eta $ in the development in Equation (8) satisfies

$$\begin{align*}\eta \geq -\dfrac{1}{2},\end{align*}$$

with equality if and only if $\hat C$ is a finite path graph.

Proof Starting from the definition of $\eta ,$ we compute the following bound:

$$ \begin{align*} \eta = -\frac{\log \alpha}{\log \tau} &= \frac{\log \mu}{\log \tau} -1 \\ &\geq \frac{\log \alpha}{\log \tau} - 1\quad\text{since }\quad\mu\geq \alpha\\ &= -\eta -1. \end{align*} $$

Solving for $\eta $ gives $\eta \geq -\frac {1}{2}.$

4.3 Case 1: X is bipartite

First, we consider the case of bipartite graphs. In this case, it must be that $\theta =2$ , since any complete graph on three or more vertices is not bipartite. The main result of this section is the following.

4.4 Case 2: X is not bipartite

Next, we consider the case that X is not bipartite.

Proof We assume that G is algebraic and derive a contradiction, and thus G must be differentially transcendental. If X is not bipartite, then $z=1$ is the only singularity on the boundary of the unit disc by [Reference Krön and Teufl10, p. 13]. Since G is algebraic, then $\omega $ in Theorem 8 is constant, and thus the singularities of G are contained in the set $(-\infty , -1)\cup [1, \infty )$ but we have already shown that are no singularities in $(1, \infty )$ by Lemma 14. We next show that there are also no singularities in $(-\infty , -1)$ . Since $\rho _d$ is a pole of order $1$ of d by Theorem 6, and d is increasing, we can deduce that $\lim _{z \rightarrow \rho _d^+} d(z) = -\infty $ . Using a limit argument applied to Equation (9), We can show that f has neither zero nor pole in the interval $(\rho _f, \rho _r)$ , and d has also no pole in the interval $(\rho _f, \rho _r)$ . Therefore, by the intermediate value theorem, for any $z_0 \in (-\infty , -1)$ , there is some $z' \in (\rho _f, \rho _r)$ such that $d(z') = z_0$ . Since $z'>\rho _f$ , $z'$ is a positive real number greater than 1, therefore, $z'$ is neither a pole nor a zero of G.

We substitute this into Equation (9)

$$\begin{align*}G^N(z') = \frac{P(z')}{Q(z')}= f(z')^N\frac{P(d(z'))}{Q(d(z'))}=f(z')^N\frac{P(z_0)}{Q(z_0)} = f(z')G(z_0).\end{align*}$$

We can conclude that $G(z_0)$ is finite and nonzero.

Thus, given the singular expansion of G at 1, we have that Q has a single zero and it is at 1, and is of order $-N\eta $ . Combining prior results:

$$\begin{align*}N\leq \deg Q = -N\eta \implies \eta \leq -1. \end{align*}$$

This directly contradicts that $\eta \geq -\frac {1}{2}$ . Thus, G cannot be algebraic.

Remark that for $\theta>2$ , it suffices to find a similarly incompatible lower bound on $\eta $ . This same proof should apply.

5.1 Discrete harmonic functions on graphs

Let $\hat C$ be a cell of a symmetrically self-similar graph with $\theta = 2$ . For $v \in V(\hat C)$ , let $H(v)$ be the probability that a random walk starting at v will reach $v_1$ before $v_2$ . Hence, $H(v_1) = 1$ and $H(v_2) = 0$ . By decomposing the first step of the random walk, we have

$$\begin{align*}H(v) = \dfrac{1}{\deg u} \displaystyle \sum_{u \in N(v)} H(u).\end{align*}$$

Then, H is a harmonic functionFootnote ³ of the graph C with boundary conditions given by the values on $v_1$ and $v_2$ . It is a classic result that there is a unique harmonic function on a graph with fixed boundary conditions. Since $\theta = 2$ , $v_1$ has only one neighbor, call it w. Then, $\alpha -1$ is the average time that a random walk starting at $v_1$ makes the step $w \rightarrow v_1$ . After that step, the only possible step is the step $v_1 \rightarrow w$ . A random walk on $\hat C$ starting at $v_1$ can have a unique decompositionFootnote ⁴ on the steps from $v_1$ to w (which we denote $v_1 \rightarrow w$ ):

$$\begin{align*}\mathcal{W}_1 = (v_1 \rightarrow w) \times \mathbf{SEQ}(\mathcal{W}_2 \times (v_1 \rightarrow w)) \times \mathcal{W}_3.\end{align*}$$

Here,

• • $\mathcal {W}_1$ is the set of walks starting at $v_1$ that stop after reaching $v_2$ , but can return to $v_1$ multiple times.

• • $\mathcal {W}_2$ is the set of walks starting at w that never reach $v_2$ and stop after reaching $v_1$ .

• • $\mathcal {W}_3$ is the set of walks starting at w that never reach $v_1$ and stop after reaching $v_2$ .

We denote by $\alpha $ the expected number of visits to $v_1$ in a walk in $\mathcal {W}_1$ (including the start of the walk). From the decomposition, we see that this is one more than the expected number of $\mathcal {W}_2$ sub-walks, as each one ends in a unique return to $v_1$ .

Let $\mathcal {W}_4$ be the set of walks starting at w that stop the first time they reach either $v_1$ or $v_2$ . We can decompose this set by endpoint ( $v_1$ and $v_2$ , respectively) into disjoint sets:

$$\begin{align*}\mathcal{W}_4 = \mathcal{W}_2 + \mathcal{W}_3.\end{align*}$$

The probability that a walk in $\mathcal {W}_4$ is also in $\mathcal {W}_2$ is precisely $H(w)$ . Therefore, the number of $\mathcal {W}_2$ in the decomposition of walk in $\mathcal {W}_1$ follows a geometric law of parameter $H(w)$ . Hence, $\alpha -1 = \dfrac {1}{1-H(w)}$ . Finding an upper bound of $\alpha $ is therefore equivalent to finding an upper bound of $H(w)$ .

5.2 The finite path graph

We can determine $\alpha $ explicitly for a path L on $\mu $ edges, as the harmonic function $H_L(v)$ is straightforward to compute. Remark that $d(v_2,v_1)=\mu $ in this case. Classically,

(11) $$ \begin{align} H_L(v) = \frac{d(v_2,v)}{d(v_2,v_1)}. \end{align} $$

Remark that $H_L(v_1) = 1$ , $H_L(v_2) = 0$ and for the vertex v at distance d from $v_2$ , $d\in \{1, \dots , d(v_1,v_2)-1 \}$ ,

$$\begin{align*}H_L(v) = \frac12\left(\frac{d-1}{d(v_1, v_2)}+\frac{d+1}{d(v_1, v_2)}\right)=\frac{d}{d(v_1, v_2)}.\end{align*}$$

Therefore, in the case of a path,

$$\begin{align*}\alpha = 1+\frac{1}{1-H_L(w)}=1+ \frac{1}{1 - \frac{d(v_1,v_2) -1}{d(v_1,v_2)}} = d(v_1,v_2) = \mu.\end{align*}$$

5.3 General case

Consider a cell graph $\hat C$ . We compare values of H, the harmonic function defined on $\hat C$ and boundary values $H(v_1)=1, H(v_2)=0$ to $H_L$ as defined in Equation (11).

First, consider the following lemma.

Lemma 20 Let $\hat C$ be a symmetric graph with $\theta = 2$ and F be the harmonic function defined on $\hat C$ the value on the boundary: $H(v_1) = 1$ and $H(v_2)=0$ . Let $u \in V(C)$ be a vertex such that there is a connected component $\tilde {C}$ of, $C \setminus \{u\}$ which contains both $v_1$ and $v_2$ . Let $\tilde {H}$ be the harmonic function defined on $\tilde {C}$ by the same boundaries as for F. Then,

$$\begin{align*}\forall x \in V(\tilde{C}), \quad H(u) \leq H(x) \Longrightarrow H(x) \leq \tilde{F}(x).\end{align*}$$

That is, if the probability of a reaching $v_1$ before $v_2$ is greater starting at x than at u, this probability is bounded above the analogous probability in the graph where u is removed, provided that $v_1, v_2$ , and x are all in the same component after the removal.

Proof The walks in $\hat C$ starting at x absorbed by a boundary can be decomposed into three disjoint sets, according to which point $v_1, v_2, u$ it will reach first. We assign the probabilities, $a,b,1-a-b$ , to be the respective probabilities that a walk starting at x will reach $v_1, v_2$ , or u first. Then, by linearity of expectation,

(12) $$ \begin{align} H(x) &= aH(v_1) + bH(v_2) + (1-a-b)H(u) \end{align} $$

(13) $$ \begin{align} &= a + (1-a-b)H(u) \end{align} $$

(14) $$ \begin{align} \implies H(u)&=\frac{H(x)-a}{1-a-b}.\end{align} $$

Let A be the set of walks that reach $v_1$ before u, and let B be the set of walks that never touch u. Then, $a=P(A)=P(A|B)P(B)=\tilde H(x) (a+b)$ since the set of walks in $\hat C$ that start at x and never touch u are in bijection with the set of walks that start at x in $\tilde {C}$ . Therefore, $\tilde H(x)=\frac {a}{a+b}$ , and if $H(u)\leq H(x)$ ,

$$ \begin{align*} \frac{H(x)-a}{1-a-b}=H(u)&\leq H(x) \implies H(x)\leq \frac{a}{a+b}\\ \implies H(x) &\leq \tilde H(x).\\[-36pt] \end{align*} $$

We can now prove the main result of the section.