Geodesic Growth of Numbered Graph Products

In this paper, we study geodesic growth of numbered graph products; these are a generalization of right-angled Coxeter groups, defined as graph products of finite cyclic groups. We first define a graph-theoretic condition called link-regularity, as well as a natural equivalence amongst link-regular numbered graphs, and show that numbered graph products associated to link-regular numbered graphs must have the same geodesic growth series. Next, we derive a formula for the geodesic growth of right-angled Coxeter groups associated to link-regular graphs. Finally, we find a system of equations that can be used to solve for the geodesic growth of numbered graph products corresponding to link-regular numbered graphs that contain no triangles and have constant vertex numbering.


Introduction
Given a group G and a generating set S, one can define the standard growth series of (G, S) to be the generating function whose nth coefficient counts the elements of G with word length n. The standard growth series of G reflects a number of important properties of G itself; a celebrated result in this area is Gromov's polynomial growth theorem which states that groups with growth series of polynomial order are virtually nilpotent [5]. Similarly, one can study the geodesic growth series of (G, S), which is the generating function whose nth coefficient counts the number of length n geodesics, or geodesic paths of length n in the Cayley graph of (G, S) starting at the identity.
A numbered graph product (NGP) is a certain group defined from a numbered graph. A numbered graph is a pair (Γ, N ) where Γ is a simplicial graph and N : V Γ → N is a map that assigns a natural number greater than 1 to each vertex of Γ. The NGP associated to (Γ, N ), denoted by NGP(Γ, N ) or just NGP(Γ), is the group with presentation v ∈ V Γ v N (v) = 1 for all v ∈ V Γ, uv = vu for all {u, v} ∈ EΓ .
We call the set S = V Γ ∪ (V Γ) −1 the standard generating set of NGP(Γ). A right-angled Coxeter group (RACG) is an NGP where N (v) = 2 for all v ∈ V Γ.
Throughout this paper, we will be interested in the geodesic growth series of NGPs associated to numbered graphs satisfying a strong combinatorial property known as linkregularity. Our first result builds on the work of [1]. There, the authors show that the geodesic growth of a RACG associated to a graph Γ is fully determined by combinatorial properties of Γ related to link-regularity. This result was then used to find distinct RACGs with the same geodesic growth.
In Section 5, we extend this result to NGPs, when the associated numbered graph satisfies a generalized definition of link-regularity: Definition 5.3. Let (Γ, N ) be a numbered graph. If U ⊆ V Γ, we let N (U ) denote the multiset {N (v) | v ∈ U }. We say Γ is link-regular if, whenever N (σ 1 ) = N (σ 2 ) for cliques σ 1 , σ 2 , we have N (Lk(σ 1 )) = N (Lk(σ 2 )). (Here, Lk(σ) denotes the link of the clique σ, defined as Lk(σ) = {v ∈ V Γ \ σ | {v} ∪ σ is a clique}, cf. Definition 4.7.) This definition is equivalent to the definition of link-regularity in [1] when the NGP in question is a RACG. We use this condition to define a certain equivalence (see Definition 5.5) on link-regular numbered graphs. The main result of Section 5 is the following: Theorem 5.14. If (Γ, N ) and (Γ , N ) are equivalent link-regular numbered graphs, then NGP(Γ) and NGP(Γ ) have the same geodesic growth series.
The equivalence of link-regular numbered graphs is weaker than the isomorphism, allowing us to find infinitely many distinct graphs with identical geodesic growth series. We prove Theorem 5.14 by sequentially generalizing the arguments from [1] to allow for the use of multisets instead of a single number.
Our next result builds on Section 5 of [1], where the authors find an explicit formula for the geodesic growth series of a RACG associated to a link-regular, triangle-free graph Γ. The paper [2] improves upon this result by allowing Γ to be tetrahedron-free. In Section 6, we further strengthen this result by obtaining the geodesic growth series of a RACG based on any link-regular graph Γ. In particular, we have the following theorem: Vol. 14:2 GEODESIC GROWTH OF NUMBERED GRAPH PRODUCTS 2:3 Theorem 6.4. Let Γ be a link-regular simplicial graph with maximum clique size d. Let 0 = |V (Γ)|, and for 1 ≤ k ≤ d, let k = |Lk(σ)|, where σ is any k-clique (this is well-defined by link-regularity of Γ). For integers 0 ≤ m ≤ d and 0 ≤ k ≤ m, set and for integers 0 ≤ i ≤ d and 0 ≤ j ≤ i, set Then the geodesic growth of RACG(Γ) is given by the rational function To prove Theorem 6.4, we apply Theorem 3.8, a method due to Chomsky and Schützenberger to the grammar that generates the geodesic words of G. This yields a large system of equations that can be solved to obtain the geodesic growth series of G. Using the fact that Γ is link-regular, we apply combinatorial arguments to obtain a simpler system of d + 1 equations, where d is the maximal clique size of Γ. Finally, we rearrange this system into a polynomial recurrence relation, which we solve via induction.
Lastly, in Section 7, we focus on NGPs when (Γ, N ) is a triangle-free, link-regular numbered graph such that the vertex numbering N is constant. We obtain the following system of equations, which can be used to solve for the geodesic growth of NGP(Γ, N ): Theorem 7.1. Let Γ be an L-regular (i.e., every vertex in Γ has L neighbors) and trianglefree simplicial graph with n vertices. Given an integer N ≥ 2, let (Γ, N ) be the numbered graph where all vertex numbers are N . The geodesic growth series G (z) of NGP(Γ) can be determined by solving the following system of equations: for 1 ≤ k, ≤ N/2 , we have Here, G k (z) and G {k, } (z) are power series such that To derive the system, we once again apply the method in Theorem 3.8. We do not derive a closed-form formula for the geodesic growth, but computer software such as Sage can be used to solve the system for small enough N .
We would like to thank and acknowledge Mark Pengitore, Alec Traaseth, and Darien Farnham of the University of Virginia's Mathematics Department for their support and assistance on this paper. Also, we would like to extend our gratitude to all the organizers of the 2022 UVA Topology REU who gave us the opportunity to make this paper possible. Lastly, we acknowledge Yago Antolín for suggesting the problem explored in this paper.

Notation
We compile a list of common notation used throughout this paper: • N will denote the positive integers (so 0 ∈ N). • A group G with a distinguished generating set S is typically denoted by the pair (G, S).
• The letter Γ usually indicates a simplicial graph (that is, an undirected graph with no double edges or loops). The vertices of Γ are denoted V Γ, and the edges of Γ are denoted EΓ.
The NGP associated to some numbered graph (Γ, N ) is written NGP(Γ, N ), or just NGP(Γ). Similarly, the RACG associated to some graph Γ is written RACG(Γ). • The letter L usually indicates a language, the letter D usually indicates a finite state automaton, and the letter C usually indicates a grammar. • The letters σ, τ , and π usually indicate cliques. The letter Σ usually indicates a powered clique (cf. Definition 4.8). • The power profile (cf. Definition 5.10) of a powered clique Σ is written P (Σ). The letters P and Q usually indicate power profiles.

Regular languages
In this section, we develop the basic theory surrounding regular languages. The reader may wish to refer to a standard text such as [6] for a more in-depth discussion on these topics. Our aim will be to define the so-called growth series of a language, and to describe a powerful technique (Theorem 3.8) used to compute the growth series of a regular language.
Definition 3.1 (Words and languages). An alphabet is a non-empty finite set, whose elements we call letters. A word x over an alphabet A is a finite sequence of the elements from A , usually written in the form x = a 1 · · · a n . A sequence a i a i+1 · · · a j is called a subword of x. The empty word is denoted by ε. A set of words over A is a language over A .
The language consisting of all words over A is denoted by A * . For a nonnegative integer n, the set of all words of length n in A * is denoted by A n . The length of a word x ∈ A * is its length when viewed as a sequence, and is written |x|. The concatenation of two words x = a 1 · · · a n and y = b 1 · · · b m is the word xy = a 1 · · · a n b 1 · · · b m . Given a word x, we write x n = x · · · x for word formed by concatenating x to itself n times, with x 0 = ε.
Regular languages are languages that are simple enough to be described by the following model of computation: Definition 3.2 (Finite state automata). A finite state automaton (FSA) D is a 5-tuple (Q, A , δ, q 0 , A), where • Q is a finite set of states, • A is an alphabet called the input alphabet, Vol. 14:2 GEODESIC GROWTH OF NUMBERED GRAPH PRODUCTS 2:5 • q 0 ∈ Q is the start state, • A ⊆ Q is the set of accept states.
If we have δ(q, a) = q , we will say that D has an a-transition from q to q . Also, we define the extended transition function δ * : Q × A * → Q recursively by Given a word x ∈ A * , we say that x ends at the state δ * (q 0 , x). If x ends at an accept state, then we say D accepts the word x, else, we say D rejects the word x. The set of all strings accepted by D is called the language accepted by D.

Definition 3.3 (Regular languages).
A regular language is a language accepted by some FSA.
Another way of describing a language is through a grammar, or a set of rules that describe how words in the language are generated. • A is a finite alphabet of terminals, • R is a finite set of production rules, which are in the form A → aB or A → ε for A, B ∈ V and a ∈ A , • S ∈ V is the start variable. If x ∈ A * , A ∈ V , and A → α is a production rule, we say xA yields xα, written xA ⇒ xα (here, α, xA, and xα are viewed as strings over V A ). If α, β ∈ (V A ) * , we say α derives β if there exists a sequence α 1 , . . . , α n such that α ⇒ α 1 ⇒ · · · ⇒ α n ⇒ β. The language generated by C is the set of all words in A * that can be derived from the start symbol S. Remark 3.5. A more standard definition of a regular grammar allows for rules of the form A → a. The definition above is equivalent, and will be more convenient for our purposes.
A well-known result is that regular grammars are equivalent to FSAs, in that they both characterize the class of regular languages. The following proposition provides an explicit description of the regular grammar that generates a given regular language.
Proposition 3.6. Let L be a regular language, and let D = (Q, A , δ, q 0 , A) be a FSA accepting L . Then L is generated by some regular grammar C = (Q, A , R, q 0 ), consisting of the production rules • q → aδ(q, a) for all q ∈ Q and a ∈ A , • q → ε for all q ∈ A.
We will be most interested in the following generating function associated to a language. Definition 3.7 (Growth series). The growth series of a language L over an alphabet A is the formal power series ∞ n=0 |L ∩ A n |z n , where the coefficient of z n is the number of length n words in L .
Sections 5 through 7 will be dedicated to studying the growth series associated to the geodesic language of a group. Our primary method of computing these series will be the following important result. One of the original descriptions of this technique (in a far more general setting) is due to [3]; we include the proof for completeness. • The grammar C is unambiguous, in the sense that every word x in the language generated by C has a unique derivation from the start symbol A 1 . • Every variable A i is reachable from the start symbol A 1 , in the sense that there is some word α ∈ (V A ) * containing A i that can be derived from A 1 . For each i = 1, . . . , n, let L i be the language consisting of all words that can be derived from A i , and let A i (z) be the growth series associated to L i . Then each series A i (z) satisfies the equation Proof. Fix a variable A i . Let c j,n = |L j ∩ A n | denote the nth coefficient of the power series A j (z). Comparing coefficients, we see that proving the theorem amounts to showing that for all n ≥ 1, and Equation (3.2) is immediate: A i → ε is a production rule if and only if ε (which is the only word of length 0) is a word in L i . Next, we prove (3.1) for n ≥ 1. Consider a word x ∈ L i ∩ A n . This word must be produced by some derivation Hence, there is some rule A i → aA j in R such that x = ay for y ∈ L j ∩ A n−1 . This gives us a map Φ : Note that this map is a bijection, with inverse given by (a, y) → ay. We claim that the codomain of Φ is a disjoint union. Suppose toward a contradiction that (a, y) were in both

This derivation can then be continued as
Vol. 14:2 GEODESIC GROWTH OF NUMBERED GRAPH PRODUCTS 2:7 But this gives two distinct derivations of the word b 1 · · · b ay, contradicting the unambiguity of C. Thus, the bijectivity of Φ implies which proves (3.1).

Numbered graph products
Definition 4.1 (Graph products). Let Γ be a simplicial graph, and let When Γ is totally disconnected, a graph product over Γ is the same as a free product. When Γ is complete, a graph product over Γ is the same as a direct product. Thus, graph products generalize both these concepts. Definition 4.3 (Numbered graph products). A numbered graph (Γ, N ) consists of a simplicial graph Γ and a map N : V Γ → N. We will refer to the numbers N (v) for v ∈ V Γ as the vertex numbers of (Γ, N ). The numbered graph product (NGP) associated to the numbered graph (Γ, N ), written NGP(Γ, N ) or NGP(Γ), is the graph product of the cyclic groups The NGP associated to (Γ, N ) is the group 3 are equal, and thus are not distinct elements in S. We can write this particular NGP in a simpler form. Given a word x ∈ S * , we may use the commuting relations to shuffle the letters v ±1 1 , v 3 to the left and the letters v ±1 2 , v ±1 4 to the right. This allows us to rewrite the group element represented by x as the concatenation of a word y ∈ v 1 , v 3 and a word z ∈ v 2 , v 4 . There are no relations between v 1 and v 3 , so they generate a free product; similarly for v 2 and v 4 . Hence, we have There is a natural evaluation map ev (G,S) : S * → G that sends a word s 1 · · · s n ∈ S * to the product s 1 · · · s n ∈ G. We say x ∈ S * evaluates in G to ev (G,S) (x). A word x ∈ S * is called a geodesic of (G, S) if there is no shorter word y ∈ S * that evaluates to the same element as x. The set of all geodesics is called the geodesic language of (G, S). The geodesic growth series of (G, S) is the growth series of the geodesic language of (G, S).
Remark 4.6. A more common interpretation of a geodesic is as the shortest path in the Cayley graph of a group (G, S). These paths are in bijection with words over the symmetric generating set S ∪ S −1 . For this reason, it is crucial in the definition of a geodesic that the generating set S is symmetric. This leads to minor inconveniences in the case of numbered graph products since the natural choice of symmetric generating set for a cyclic group of order 2 contains a single element, whereas for a cyclic group of any other order the symmetric generating set contains two elements. This discrepancy will lead to important edge cases in the results to follow.
Note that whenever we speak of the geodesics (or geodesic growth) of some numbered graph product NGP(Γ), we are always working with respect to the standard generating set of NGP(Γ).
In the remainder of this section, our goal will be to construct a FSA that recognizes the geodesic language of an NGP. It is already known that this language is regular; the paper [7] constructs a FSA that recognizes the geodesic language of any graph product whose vertex groups have regular geodesic languages. To do this, they modify the FSA's associated to individual vertex groups, then show that the product of these FSA's recognize the geodesic language of the graph product. We will follow their construction, but we will interpret the product FSA in a setting specific to NGPs. In particular, the states will be objects called powered cliques. Definition 4.7 (Cliques and Links). Let Γ be a simplicial graph. A subset σ ⊂ V Γ is called a clique if the vertices of σ span a complete subgraph of Γ. If σ has i elements, we call σ an i-clique. A subset τ ⊆ σ is called a subclique of σ. We define the link of a clique to be the set where · is the floor function. We refer to the number Σ(v) as the power of Σ at the vertex v. We will often define a powered clique by first specifying its support, then defining its power at vertices in its support. The empty powered clique is the map Σ : V Γ → Z that has empty support. • The accept states are the powered cliques of (Γ, N ), and there is a single reject state q rej ∈ Q \ A. The start state is the empty powered clique.
In all other cases, we define the transition δ(Σ, v ±1 ) to be q fail .
To prove Theorem 4.9, we will need the following fact from [7] about the geodesic language of graph products. A rigorous proof of this fact involves developing a normal form for graph products, see [4] for the construction of such a normal form.
Lemma 4.10. Let G be a graph product of the groups (G v , S v ), and let S = v∈V S v . Given a word s 1 · · · s n ∈ S * , we define the following transformation, called shuffling: (S) Interchange two consecutive letters s i ∈ S u and s i+1 ∈ S v if {u, v} is an edge in Γ. Then a word x ∈ S * is not a geodesic of (G, S) if and only if there is a way to shuffle the letters of x to form a subword in S * v that is not a geodesic of (G v , S v ). Proof of Theorem 4.9. We write V Γ = {v 1 , . . . , v n }, Fix an index i, and let m := N (v i )/2 . The geodesic language of the cyclic group G i is the set of words there is a single reject state q rej , and the start state is the identity.
. Clearly, D i accepts the geodesic language of G i . Next, we modify the FSA D i to read words over the alphabet S. The modified FSA, which we call D i , is the same as D i , but has the following additional transitions: Figure 1. Observe that D i rejects a word x ∈ S * if and only if there is a way to shuffle the letters of x (as defined by Lemma 4.10) to form a subword in S * i that is not a geodesic of (G i , S i ).
Next, we construct the product D of the modified FSA's D i . The accept states of D are There is a single reject state q rej , and the tuple (1, . . . , 1) is designated as the start state. Given a generator s ∈ S, there is an Vol. 14:2 By construction, D rejects a word in S * if and only if the word is rejected by some individual FSA D i . Therefore, D rejects a word in x ∈ S * if and only if there is a way to shuffle the letters of x to form a subword in any S * i that is not a geodesic of (G i , S i ). By Lemma 4.10, D accepts the geodesic language of (G, S).
We claim that the FSA D is exactly the one described by the theorem statement. Observe that the states of D are in bijection with powered cliques, where to each state (v k 1 1 , . . . , v kn n ) we associate the powered clique v i → k i , and to each powered clique Σ we associate the state (v ). Thus, we may view the states of D as powered cliques.
It remains to show that the transition map of D is given by δ. Let Σ be a powered clique, which corresponds to the state (v

Link-regular NGPs
Now that we have introduced the necessary background, we are prepared to explore the geodesic growth of NGPs. In particular, we will focus on NGPs associated to numbered Vol. 14:2 GEODESIC GROWTH OF NUMBERED GRAPH PRODUCTS 2:11 graphs that obey a strong regularity condition called link-regularity. Our main result in this section will show that NGPs associated to link-regular numbered graphs which are equivalent (in a certain sense, which we will define) have the same geodesic growth series. This result then allows us to provide examples of NGPs associated to non-isomorphic graphs that have the same geodesic growth series. We begin by defining link-regular numbered graphs and the equivalence of such graphs. Note that when the vertex numbers N (v) are all 2, our definition of link-regularity (Definition 5.3) is equivalent to Definition 3.1 in [1]. We will discuss this case at length in Section 6. Remark 5.2. Informally, we will write multisets as sets with repeated elements, e.g., {a, a, b, c, c, c}. Similarly, we will often define multisets with set-builder notation, e.g., { n/2 | n ∈ N} is the multi-set {0, 1, 1, 2, 2, 3, 3, . . . }.   The equivalence of link-regular numbered graphs is less strict than graph isomorphism, as we showcase in the following example. Our goal is to prove that the NGPs associated to equivalent link-regular numbered graphs (Γ, N ) and (Γ , N ) have the same geodesic growth series. To do this, we will study the FSAs D and D (which were constructed in Theorem 4.9) that recognize the geodesic languages of NGP(Γ) and NGP(Γ ), respectively. Because the transitions of D and D are defined in terms of cliques and links, we will need several results comparing the cliques and links of Γ and Γ .
Proof. By definition, the disjoint union (Γ Γ , N N ) is link-regular. Now, σ and σ are subcliques of (Γ Γ , N N ) and so the claim follows by Lemma 5.8. Definition 5.10. Let Σ : V Γ → Z be a powered clique of a numbered graph (Γ, N ). The power profile of Σ, written P (Σ), is the multiset {(Σ(v), N (v)) | v ∈ supp Σ}. If the power profile of Σ is P , we say Σ is a P -state.
Lemma 5.11. Let (Γ, N ) and (Γ , N ) be equivalent link-regular numbered graphs, and let Σ : V Γ → Z and Σ : V Γ → Z be powered cliques such that P (Σ) = P (Σ ). Then there is a bijection Φ : for all v ∈ V Γ. Here, δ and δ denote the transition maps of D and D , respectively.
Corollary 5.12. Let (Γ, N ) and (Γ , N ) be equivalent link-regular numbered graphs, and let D and D be the FSAs constructed in Theorem 4.9 that recognize the geodesic language of NGP(Γ) and NGP(Γ ). Let P and Q be arbitrary power profiles of states in D and D . If Σ and Σ are P -states of D and D , respectively, then the number of transitions from Σ to a Q-state in D is the same as the number of transitions from Σ to a Q-state in D .
Proof. Let Φ : V Γ → V Γ be as in Lemma 5.11. By (5.2), δ(Σ, v) is a Q-state if and only if δ(Σ , Φ(v)) is a Q-state. Because Φ is a bijection, there is a bijective correspondence between the transitions from Σ to Q-states in D and the transitions from Σ to Q-state in D , so we are done.
Corollary 5.12 allows us to define the following quantities β P →Q , which are well-defined up to equivalence of link-regular numbered graphs.
Definition 5.13. Fix some numbered graph (Γ, N ), and let D be the FSA that recognizes the geodesic language of NGP(Γ). Given arbitrary power profiles P and Q of powered cliques corresponding to the states in D, we define β P →Q to be the number of transitions from a fixed P -state to a Q-state. Proof. Let G = NGP(Γ), and let S be the standard generating set of G. We will show that the geodesic growth series of (G, S) can be written in terms of the quantities β P →Q . Because the values β P →Q are the same for equivalent link-regular numbered graphs, this proves the theorem. Let d be the maximum clique size of Γ, and let M be the maximum number N (v) assigned to a vertex. We let P denote the collection of all multisets with elements in have at most d elements. Note that P contains all possible power profiles of states in D. For each P ∈ P, let B P (m) denote the collection of length m words in S * that end at a P -state when processed by D. Each geodesic of (G, S) belongs to a unique set B P (m), so the total number of m-length geodesic words is given by the sum Therefore, the geodesic growth series of (G, S) can be written as Hence, it suffices to show that the quantities |B P (m)| can be written in terms of the quantities β P →Q . This is trivial for the case m = 0, where |B P (m)| = 0 if P is non-empty, and |B P (m)| = 1 if P is empty. Inductively, assume there is some m > 0 such that all the |B P (m)| can be written in terms of the β P →Q . We want to show that the same is true for all the |B P (m + 1)|. Fix P ∈ P. Observe that a word x ∈ B P (m) can be written as x = ys for a word y ∈ B Q (m) with Q ∈ P and a single letter s. That is, a choice of a word x ∈ B P (m) consists of a choice of a power profile Q ∈ P, then a choice of a word y ∈ B Q (m), and finally a choice of s ∈ S for which there is an s-transition from the Q-state that y ends at to some P -state. Hence, we have the recurrence This proves that |B P (m)| can be written in terms of the quantities β P →Q , and so we are done by induction.
Using Theorem 5.14, we are able to find examples of NGPs associated to non-isomorphic graphs that have the same geodesic growth series. For example, the NGPs associated to the numbered graphs from Example 5.6 have the same geodesic growth series, despite the graphs being non-isomorphic. More generally, the NGP associated to the disjoint union of identical n-cycles has the same geodesic growth series as the NGP associated to the 2n-cycle with the same vertex numbers. Another example of this construction is the following: Example 5.15. The disjoint union of the two 5-cycles and the 10-cycle shown below are equivalent as link-regular numbered graphs, so their associated NGPs have the same geodesic growth. These examples demonstrate the following corollary to Theorem 5.14.
Corollary 5.16. There are infinitely many pairs of NGPs associated to non-isomorphic graphs that have the same geodesic growth.

Right-angled coxeter groups
We now shift our attention to a specific class of NGPs known as right-angled Coxeter groups.
Definition 6.1 (Right-angled Coxeter groups). A right-angled Coxeter group (RACG) is an NGP associated to a numbered graph whose vertex numbers are all 2. In particular, we may speak of the RACG associated to a simplicial graph Γ, written RACG(Γ), which we define to be the NGP associated to the numbered graph (Γ, N ) where N (v) = 2 for all v ∈ V (Γ). As with NGPs, whenever we refer to the geodesics (or geodesic growth) of some RACG, we are working with respect to the standard generating set.
The results we have so far derived for NGPs become much simpler in the context of RACGs. For instance, consider the FSA D we constructed in Theorem 4.9 that recognizes the geodesic language of some RACG. The states of D are powered cliques, but because every vertex number is 2, a powered clique can only assign the powers 0 and 1 to vertices. Thus, we may identify a powered clique with its support, which must be a clique. In particular, the states of D are in bijection with the cliques of the graph. Rewriting the transition map in terms of cliques, we obtain the following special case of Theorem 4.9.
As before, we can define two link-regular simplicial graphs to be equivalent if their disjoint union is link-regular. In this setting, Theorem 5.14 says that RACGs associated to equivalent link-regular simplicial graphs have identical geodesic growth series. This is one of the main results of [1]. In this section, we generalize their result by deriving the following closed-form expression for geodesic growth.
Then the geodesic growth of RACG(Γ) is given by the rational function Previously, the geodesic growth series of RACG(Γ) was only known for maximum clique size d ≤ 3, proven in [1] and [2]. Detailed calculations using Theorem 6.4, and a Sage program that automatically carries out these calculations, can be found in [8]. This program was used to compute the following formula when the maximum clique size is d = 4. Corollary 6.5. If Γ is a link-regular simplicial graph that does not contain 5-cliques, the geodesic growth of RACG(Γ) is the rational function q(z)/p(z), where q(z) = 24z 4 + (− 1 2 3 + 4 1 2 + 2 3 − 12 1 − 4 2 − 2 3 + 50)z 3 The remainder of this section is devoted to the proof of Theorem 6.4. We fix the following notation. Let Γ, d, and k be defined as in the statement of Theorem 6.4. Let S be the standard generating set of RACG(Γ). Let C denote the set of all cliques of Γ. Let C m denote the set of all ordered m-cliques of Γ, i.e., tuples (v 1 , . . . , v m ) consisting of distinct Vol. 14:2 vertices such that {v 1 , . . . , v m } is a clique. The elements of C m will be denotedσ, where σ is the underlying unordered clique. We begin by using Theorem 3.8 to obtain a system of equations that allows us to solve for the geodesic growth of RACG(Γ). Applying Proposition 3.6 to the FSA D in Proposition 6.2, we obtain the following grammar that accepts the geodesic language of RACG(Γ): to each clique σ ∈ C , we associate the grammar variable ∆ σ , and the production rules The start symbol is the variable ∆ ∅ . Observe that the vertices not in σ can be counted by first choosing a (possibly empty) subset τ ⊆ σ, and then choosing a vertex in Lk(σ; τ ); more precisely, we have Hence, the above grammar rules can be equivalently written To be able to apply Theorem 3.8, we first need to check two conditions: Lemma 6.6. The grammar in (6.1) is unambiguous, and every variable is reachable.
Proof. We first show that (6.1) is unambiguous. Observe that, given any grammar variable ∆ σ and some vertex u ∈ V Γ, there is at most one rule ∆ σ → α where the right-hand side α contains u. This implies that, if the grammar generates some word x = v 1 · · · v n ∈ S * , then there is only one possible production rule that could have been applied at each step. Hence, x has a unique derivation from the start symbol, as required.
For the second claim, fix a variable ∆ σ for some clique σ = {v 1 , . . . , v m }. Then we have the derivation which proves that ∆ σ is reachable. Now, we may apply Theorem 3.8 to obtain the equations for all σ ∈ C , where ∆ σ (z) denotes the growth series of the language consisting of all words that can be derived from ∆ σ . To simplify this system, we introduce the power series We want to convert the system of equations (6.2), which is written in terms of the power series ∆ σ (z), into a system of equations written in terms of the power series G m (z). The following proposition is the first step in doing so. The geodesic growth series of RACG(Γ) is then given by G 0 (z).
Given an ordered clique (v 1 , . . . , v k , u) ∈ C k+1 , we define the power series ∆ (v 1 ,...,v k ,u) (z) to equal the power series ∆ {v 1 ,...,v k ,u} (z). We do this to improve notation in order to keep track of the ordering for Proposition 6.8. Making use of this change converts our system into the equivalent form is an integer N m,k that is independent of the vertices v 1 , . . . , v k , u. Furthermore, the integers N m,k satisfy the following properties: • N m,m = 1.
In Case 1 and Case 2, we have shown that the base cases of this recurrence are independent ofτ and u. Therefore, N m,k (τ , u) is independent ofτ and u. Writing N m,k = N m,k (τ , u), we obtain the required formula.
Therefore, by Theorem 6.8, we see that our system in (6.6) becomes we are left with the system of equations: The clear next step is to bring the recurrence N m,k into a closed form.
Lemma 6.9. For 1 ≤ m ≤ d and 0 ≤ k ≤ m, the recurrence N m,k in Prop. 6.8 has the closed form solution In particular, we see that the closed form holds for N 1,0 , N 1,1 , N 2,0 , N 2,1 , and N 2,2 . Now, we proceed by induction on m. Fix m ≥ 3 and suppose that the closed form holds for N m,k for all 0 ≤ k ≤ m. We induct backward on k to establish the statement for N m+1,k for all 0 ≤ k ≤ m + 1. By the preceding paragraph, we know that for k ≥ m − 1, the closed form holds. Thus, we fix k < m − 1, and suppose that the closed form holds for N m+1,k+1 . We will show the closed form is then correct for N m+1,k . We have The product in (6.9) agrees with the closed form solution, so we need only argue that the difference between ( * ) and ( * * ) is equal to the sum in the closed form. We rewrite ( * ) as Taking the difference of (6.10) and (6.11) and applying Pascal's rule for binomial coefficients yields As Thus, the closed form holds for N m+1,k for all 0 ≤ k ≤ m + 1. By induction on m, the closed form holds for all N m,k .
We will now obtain a solution to the system in (6.8) in terms of a polynomial recurrence. First, set a 0 = 1 and a m = 0 · · · m−1 for all 1 ≤ m ≤ d. b m,k G k+1 (z) (6.12)