**1. Introduction**

One of the best-known connections between groups and graph theory was presented by A. Cayley [1]. He gave a group *G* as a directed graph, where the vertices correspond to elements of *G* and the edges to multiplication by group generators and their inverses. Such a graph is called a Cayley diagram or Cayley graph of *G*. It is a central tool in combinatorial and geometric group theory.

Recent works reveal many different ways of associating a graph to a given finite group, most of which were inspired by a question posed by P. Erdös [2]. These differences lie in the adjacency criterion used to relate two group elements constituting the set of vertices of such a graph. Some essential authors in this context are A. Abdollahi [3], A. Ballester-Bolinches et al. [4–8], A. Lucchini [9,10], and D. Hai-Reuven [11], among others.

Our notation will be standard, as in [12] and [13] for groups and graphs. Let *G* = &*g*1, ... , *gn*' be a finitely generated group and suppose now that every element *g* ∈ *G* can be uniquely written as follows

$$\mathfrak{g} = \prod\_{i=1}^{n} \mathfrak{g}\_i^{c\_i},\tag{1}$$

with 0 ≤ *<sup>i</sup>* < *mi*, and 1 ≤ *i* ≤ *n*. The numbers *mi* can be, for example, the orders of the corresponding elements in the finite case, but they may also differ from these orders.

To determine a measure of the separation between two elements of *G*, we introduce the following distance map *d*<sup>1</sup> : *G* × *G* −→ N0, defined by

$$d\_1(\emptyset, h) = d\_1\left(\prod\_{i=1}^n \mathbb{g}\_i^{\varepsilon\_i} \prod\_{i=1}^n \mathbb{g}\_i^{\delta\_i}\right) = \sum\_{i=1}^n |\epsilon\_i - \delta\_i|. \tag{2}$$

The set *G* endowed with this distance *d* is a metric space. Note that *d*<sup>1</sup> is just the Minkowski *lp* metric for *p* = 1 in {(1, ... , *n*) | 0 ≤ *<sup>i</sup>* < *mi*}. This is also called the taxicab distance, Manhattan distance, or grid distance.

G. Diaz-Porto and A. Torres-Grandisson introduced *t*-graphs using Minkowski's metric in [14,15]. These graphs can be defined by the group *G* as the underlying set of

**Citation:** Diaz-Porto, G.; Gutierrez, I.; Torres-Grandisson, A. The *t*-Graphs over Finitely Generated Groups and the Minkowski Metric. *Mathematics* **2022**, *10*, 3030. https://doi.org/ 10.3390/math10173030

Academic Editors: Hashem Bordbar and Irina Cristea

Received: 13 July 2022 Accepted: 5 August 2022 Published: 23 August 2022

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vertices and the following adjacency criteria: Let *t* be an integer number with 1 ≤ *t* ≤ *n*. We say that *g*, *h* ∈ *G* are adjacent if and only if *d*1(*g*, *h*) = *t*.

The simplest example is when *G* is a finite cyclic group. Let *G* = &*g*' be a cyclic group with the finite order *m*. That is, *G* = {1, *g*, ··· , *gm*−1}. From (2), we have

$$d\_1(\mathbb{g}^i, \mathbb{g}^j) = |i - j|\_\prime \text{ for all } 0 \le i, j \le m - 1. \tag{3}$$

This means that in the *t*-graph of *G* there exists an edge between *g<sup>i</sup>* and *g<sup>j</sup>* if and only if |*i* − *j*| = *t*. Defining on *G* the following relation

$$\text{g}^i \sim \text{g}^j \iff i \equiv j \bmod t,\tag{4}$$

where ∼ is an equivalence relation, and then we have a partition of *G* in *t* classes given by

$$\{\lg^i\} := \{\lg^j \in G \mid j \equiv i \bmod t\},\tag{5}$$

where *i* ∈ {0, 1, ... , *t* − 1}. Then, the *t*-graph of a finite cyclic group *G* can be viewed as the union of *t* connected components, consisting of path graphs or isolated points. Consequently for *t* ≥ 2, the *t*-graph is non-connected and 2-chromatic. The 1-graph of *G* is a finite path graph and then connected.

If *t* is a divisor of the group order *m*, then it is well known that *G* has a cyclic subgroup *U* of order *m*/*t*, and the elements of *U* form a subgraph with *m*/*n* vertices, which is a connected component of the *t*-graph of *G*.

If *G* = &*g*' is an infinite cyclic group, then the 1-graph of G is an infinite path graph. This statement follows directly from the definition of the *t*-graph.

An immediate consequence of the above discussion is that if *G* is a finite abelian group, say *G* = &*g*1'×···×&*gn*', with ord(*gj*) = *j*. Then, the 1-graph of *G* is the Cartesian product of *n* path graphs of lengths *j*, respectively. That is an *n*-dimensional square grid graph. In general, using the above example, the *t*-graph of *G* is the Cartesian product of *t* components.

In the general case, if *G* is a direct product of cyclic groups with at least one infinite factor, then the *t*-graph of *G* is an infinite rectangular grid graph.

The first thing we can observe is that for a group *G* different generating systems can give different graphs. For instance, the groups Z<sup>4</sup> × Z<sup>6</sup> and Z<sup>2</sup> × Z<sup>12</sup> are isomorphic, but the graphs associated with the natural generating sets corresponding to these ways to present the group *G* are different.

On the other hand, if two groups admit generating systems such that every element *g* can be described as in (1), then it is possible that the corresponding *t*-graphs are the same, even though the groups are not isomorphic. We can see this in the following example. It is well known that the dihedral group *Dn* and the quaternion group *Q*<sup>8</sup> have the subsequent group presentation, respectively,

$$D\_n = \langle a, b \mid a^2 = b^n = 1, \ aba = b^{-1} \rangle,\tag{6}$$

$$Q\_8 = \langle a, b \mid a^4 = 1, \ a^2 = b^2, \ bab^{-1} = a^{-1} \rangle. \tag{7}$$

Furthermore,

$$\mathbb{Z}\_2 \times \mathbb{Z}\_4 = \langle a, b \mid a^2 = b^4 = 1, \ ab = ba \rangle. \tag{8}$$

Note that, in terms of their generators, the elements of *D*4, *Q*8, and Z<sup>2</sup> × Z<sup>4</sup> can be written as follows

{1, *a*, *b*, *b*2, *b*3, *ab*, *ab*2, *ab*3}. (9)

This means that the three groups have the same distance table (see Table 1) and, consequently, the same *t*-graphs for all *t*.


**Table 1.** Table of distances of Z<sup>2</sup> × Z4, *D*4, and *Q*8.

An illustration of the first four *t*-graphs of these three groups is presented in the following Figure 1.

**Figure 1.** Some *t*-graphs of Z<sup>2</sup> × Z4, *D*4, and *Q*8.

Despite being non-isomorphic groups, these groups have precisely the same *t*-graphs since the metric used to define the adjacency criterion only considers the writing of the group's elements and not how they interact. This leads to the conclusion that any twogenerator finite group *G* = &*a*, *b*', in which every element can be written in the form *a<sup>i</sup> bj* with 0 ≤ *i* ≤ ord(*a*) − 1 and 0 ≤ *j* ≤ ord(*b*) − 1, has the same *t*-graphs as the group Zord(*a*) × Zord(*b*) since, when considering the form in which its elements are written in terms of the generators, the underlying sets are the same.

Therefore, to study the *t*-graphs of a finite group *G*, it is sufficient to consider abelian groups, expressed as products of cyclic groups. Naturally, this implies asking oneself, given an arbitrary group *G*, how to determine the abelian group with which it will share the same *t*-graphs. For example, the symmetric group of degree five has the same *t*-graph as Z<sup>2</sup> × Z<sup>3</sup> × Z<sup>4</sup> × Z5. In fact, in general, the group Sym(*n*) can be factorized in the form Sym(*n*) = Sym(*n* − 1)&(12 ··· *n*)', and, applying this property inductively, we have that Sym(*n*) is generated by the set {(12),(123), ··· ,(12 ··· , *n*)}. In particular, the set

$$\{(12)^{l}(123)^{l}(1234)^{k}(12345)^{l} \mid i = 0, 1, j = 0, 1, 2, k = 0, 1, 2, 3, l = 0, 1, 2, 3, 4\}$$

is exactly Sym(5).

On the other hand, this situation brings the possibility of studying *t*-graphs by defining the adjacency criterion in terms of another metric. This change may imply that the group structure plays a more critical role.

The main goal of this paper is to obtain some characterizations of the *t*-graphs G associated with the two-generator finite group *G* that can be expressed in the form

$$G = \langle a, b \rangle = \{ a^i b^j \mid 0 \le i \le m, \ 0 \le j \le n \}. \tag{10}$$

where *m* ≤ ord(*a*) y *n* ≤ ord(*b*); *n*, *m* ∈ Z. These numbers *m* and *n* depend exclusively on the structure, namely on the group's presentation and the order of *G*. We determine the number of connected components of G depending on whether *t* is an even or odd number.

## **2. Preliminaries on** *t***-Graphs**

A desirable property of the *t*-graphs is that every subgroup *H* of a group *G* naturally results in a subgraph. However, this is, in general, not true. For example, let *G* be the Klein four-group, say *G* = {1, *a*, *b*, *ab*} and *H* = {1, *ab*}. Concerning their natural generating systems, *ab* and one are not adjacent in the 1-graph of *G*. Nevertheless, in *H*, they are adjacent.

**Lemma 1.** *Let G* = &*g*1, ... , *gn*' *be a finitely generated group and H* ≤ *G, with H* = &*h*1, ... , *hn*' *and hj* = *g kj <sup>j</sup> for some natural numbers kj. Then, the t-graph of H is a subgraph of the t-graph of G.*

**Proof.** It follows immediately from the definition of the *t*-graph.

**Lemma 2.** *Let G* = &*g*1, ··· , *gn*' *be a finitely generated group and suppose now that every element in g* ∈ *G can be uniquely written as g* = ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *<sup>g</sup><sup>i</sup> <sup>i</sup> with* 0 ≤ *<sup>i</sup>* < *mi and* 1 ≤ *i* ≤ *n. Further, let H* = &*h*1, ··· , *hn*' *be a finitely generated group with the same property. If G and H are isomorphic, then the corresponding t-graphs are isomorphic, for all natural numbers t.*

**Proof.** Let *f* : *G* −→ *H* be a group isomorphism with *f*(*gi*) = *hi*, and let G = (*G*, *E*1) and H = (*H*, *E*2) be the corresponding *t*-graphs of *G* and *H*, respectively. Suppose that {*x*, *y*} ∈ *E*<sup>1</sup> with *x* = ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *<sup>g</sup><sup>i</sup> <sup>i</sup>* and *<sup>y</sup>* <sup>=</sup> <sup>∏</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *g δi <sup>i</sup>* . Then, *d*1(*x*, *y*) = *t*, and we have

$$\begin{aligned} d\_1(f(\mathbf{x}), f(\mathbf{y})) &= d\_1(\prod\_{i=1}^n f(\mathcal{g}\_i)^{c\_i}, \prod\_{i=1}^n f(\mathcal{g}\_i)^{\delta\_i}) = d\_1 \left( \prod\_{i=1}^n h\_i^{c\_i}, \prod\_{i=1}^n h\_i^{\delta\_i} \right), \\ &= \sum\_{i=1}^n |\varepsilon\_i - \delta\_i| = d\_1(\mathbf{x}, \mathbf{y}). \end{aligned}$$

It follows that { *f*(*x*), *f*(*y*)} ∈ *E*2.

**Remark 1.** *Note that the reciprocal of the statement in Lemma 2 is, in general, not true. For example, the t-graphs of the dihedral D*<sup>4</sup> *and the quaternions group, Q*<sup>8</sup> *are isomorphic even though D*<sup>4</sup> ∼= *<sup>Q</sup>*8*.*

To study *t*-graphs in the given context, we can use the spectral theory of graphs, which consists of studying the properties of the Laplacian matrix of a graph, more specifically, its eigenvalues and eigenvectors.

The Laplacian matrix of G = (*V*, *E*) is the *n* × *n* matrix *L* = (*lij*) indexed by *V*, whose (*i*, *j*)-entry is defined as follows

$$l\_{ij} = \begin{cases} -1 & \text{if } \{v\_i, v\_j\} \in E\\ \text{deg}(v\_i) & \text{if } i = j\\ 0 & \text{otherwise.} \end{cases} \tag{11}$$

To analyze the behavior of the number of connected components *k*(G) of the *t*graphs defined on a group *G*, we use the following theorem, which allows us to realize Tables 2 and 3. A proof of this theorem can be found in [16] (Theorem 7.1).

**Theorem 1.** *A graph* G *has k connected components if and only if the algebraic multiplicity of zero as the Laplacian eigenvalue is k.*

In the following, to study the *t*-graphs associated with a finite group *G*, we will consider only finite two-generator groups, which can be expressed in the form (10). These numbers *m* and *n* depend exclusively on the structure, namely on the group's presentation and the order of *G*.

Let *G* be such a group. To observe the behavior of the number of connected components *k*(G) of a *t*-graph G determined by a group *G*, we make use of Theorem 1, with which we were able to make the following tables:


**Table 2.** Number of connected components of the *t*-graphs on Z*<sup>n</sup>* × Z2.

**Table 3.** Number of connected components of the *t*-graphs on Z*<sup>n</sup>* × Z3.


**Remark 2.** *Note in the previous tables that k*(G) *has the same value up to a certain value of t where, if <sup>t</sup> is even, <sup>k</sup>*(G) = <sup>2</sup>*, and, if <sup>t</sup> is odd, then <sup>k</sup>*(G) = <sup>1</sup> *and, when <sup>t</sup>* <sup>&</sup>gt; / *<sup>m</sup>*+*n*−<sup>2</sup> <sup>2</sup> 0 *, then k*(G) *has a value with the following possible pattern:*


This fact leads us to state the first theorem in the next section, which allows us to characterize first the *t*-graphs associated with two-generator groups in the form (10), concerning the number of connected components.

#### **3. The** *t***-Graph of Some Two-Generator Groups**

This section considers the *t*-graph of a particular case of two-generator groups. Specifically, we suppose that *a* is an involution and *b* has an order *n*. For example, the group *G* can be the abelian group Z<sup>2</sup> × Z*<sup>n</sup>* or the dihedral group *Dn* of order *n*.

**Lemma 3.** *Let G be a two-generator group in the form* (10) *with n*, *m* ≥ 2*, and* G *is the corresponding t-graph of G. Then,* <sup>G</sup> *has no isolated points if and only if t* <sup>≤</sup> / *<sup>m</sup>*+*n*−<sup>2</sup> <sup>2</sup> 0 *.*

**Proof.** Let *x* = *a<sup>i</sup> bj* , *y* = *akbl* ∈ *G* with

$$d\_1(x, y) = |i - k| + |j - l| = t. \tag{12}$$

Then, *t* ∈ {0, ... , *m* + *n* − 2}, and suppose |*i* − *k*| = *s* ∈ {0, ... , *m* − 1}. This implies that |*j* − *l*| = *t* − *s* ∈ {0, ... , *n* − 1}. Note that if *t* − *s* > *n* − 1, the equality (12) is not verified. That is, there is no edge between *x* and *y*. Then, in order not to have isolated points, it must be fulfilled that *t* − *s* ≤ *n* − 1 with *s* ∈ {0, ... , *m* − 1}. Moreover, *t* ≤ *n* − 1. Analogously, it follows that *t* ≤ *m* − 1. Consequently, 2*t* ≤ *m* + *n* − 2, and, therefore, *<sup>t</sup>* <sup>≤</sup> / *<sup>m</sup>*+*n*−<sup>2</sup> <sup>2</sup> 0 .

**Theorem 2.** *Let G be a two-generator group in the form* (10) *with n*, *m* ≥ 2*, and* G = (*G*, *E*) *be the corresponding t-graph with t* <sup>≤</sup> / *<sup>m</sup>*+*n*−<sup>2</sup> <sup>2</sup> 0 *.*


**Proof.** From the above lemma, we have that the condition *<sup>t</sup>* <sup>≤</sup> / *<sup>m</sup>*+*n*−<sup>2</sup> <sup>2</sup> 0 implies that G has no isolated points. We now differentiate two possible cases.

1. Let *t* be an even number. We define C<sup>1</sup> = (*V*1, *E*1) and C<sup>2</sup> = (*V*2, *E*2), the subgraph of G, as follows:

$$\mathcal{V}\_1 := \{ a^i b^j \mid i + j \equiv 0 \bmod 2 \}, \tag{13}$$

$$E\_1 := \{ \{a^ib^j, a^kb^l\} \mid i+j, k+l \equiv 0 \bmod 2 \land \ |i-k|+|j-l|=t \},\tag{14}$$

and

$$\mathcal{V}\_2 := \{ a^i b^j \mid i+j \equiv 1 \bmod 2 \},\tag{15}$$

$$E\_2 := \{ \{a^ib^j, a^kb^l\} \mid i+j, k+l \equiv 1 \bmod 2 \land \ |i-k|+|j-l|=t \}. \tag{16}$$

It is clear that *V*<sup>1</sup> ∪ *V*<sup>2</sup> = *G*, and then *k*(G) = 2.

2. Let *t* be an even number, and *x* = *a<sup>i</sup> b<sup>j</sup>* ∈ *G* be arbitrary. If *i* + *j* ≡ 1 mod 2, then we consider the sets

$$\{a^k b^l \mid i, k+l \equiv 0 \bmod 2, j \equiv 1 \bmod 2 \land |i-k| + |j-l| = t\} \tag{17}$$

$$\{a^k b^l \mid j, k+l \equiv 0 \bmod 2, i \equiv 1 \bmod 2 \land \ |i-k|+|j-l|=t\}.\tag{18}$$

Since G has no isolated points, at least one of these sets is non-empty, and then {*a<sup>i</sup> bj* , *akbl* } ∈ *E*.

If *i* + *j* ≡ 0 mod 2, then a similar analysis leads to the same conclusion. Then, we have that G is a connected graph.

The next theorem shows that the 1-graph associated with a finite dihedral group *Dn* has a simple structure. It corresponds to a square (*n* × 2)-grid, as shown in Figure 2 below. Therefore, this graph is bichromatic or bipartite.

**Theorem 3.** *The* 1*-graph of Dn is bipartite.*

**Proof.** From (6), we have that

$$D\_n = \{1, b, \dots, b^{n-1}\} \cup \{ab, \dots, ab^{n-1}\}.\tag{19}$$

Note that

$$d\_1(b^i, b^{i+1}) = d\_1(ab^i, ab^{i+1}) = 1,\tag{20}$$

then, the sets {1, *b*, ··· , *bn*−1} and {*a*, *ab*, ··· , *abn*−1} form a bipartition of the vertex set *Dn*.

**Figure 2.** The 1-graph of *Dn*.

Theorem 2 leads to a complete characterization of the *t*-graphs associated with *Dn*. However, before characterizing the *t*-graphs on dihedral groups, let us first look at some useful lemmas.

**Lemma 4.** *Let* G = (*Dn*, *E*) *be the t-graph of Dn. Then,*

$$|E| = \begin{cases} 4(n-t) + 2 & \text{if } t > 1 \\ 3n - 2 & \text{if } t = 1. \end{cases} \tag{21}$$

**Proof.** Let *x* = *a<sup>i</sup> bj* , *y* = *akbl* ∈ *Dn*, then, 0 ≤ *i*, *k* ≤ 1 and 0 ≤ *j*, *l* ≤ *n* − 1. If *d*1(*x*, *y*) = |*i* − *k*| + |*j* − *l*| = *t*, then, for |*i* − *k*|, we have the following cases:


If *t* > 1, then there are 2(*n* − *t*) + 2(*n* − *t* + 1) ways of constructing an edge between two elements of *Dn*. Therefore, we have that |*E*| = 4(*n* − *t*) + 2.

If *t* = 1 then we the same argument we have that |*E*| = 3*n* − 2.

**Lemma 5.** *Let f* : *Dn* −→ *Dn be defined as follows*

$$f(a^ib^j) = \begin{cases} b^j & \text{If } i = 1\\ ab^j & \text{If } i = 0. \end{cases} \tag{22}$$

*Then, f is an isometry under the Minkowski metric* (2)*. Further, if we restrict f to U* ⊂ *Dn, we have that U and f*(*U*) *are also isometric under the Minkowski metric.*

**Proof.** It is immediate that *f* is an injective function and (*f* ◦ *f*)(*x*) = *x*, for all *x* ∈ *Dn*. That is, *f* is bijective. To prove that *f* is an isometry, let *a<sup>i</sup> bj* , *akbl* ∈ *Dn*. Then,


Therefore, *f* is an isometry on *Dn*. The other statement is clear.
