**Theorem 4.** *(Characterization of t-graphs on Dn)*

*Let* G = (*Dn*, *E*) *the t-graph of Dn with n* ≥ 2*. We define r* := / *<sup>n</sup>* 2 0 *.*


$$k(\mathcal{G}) = \begin{cases} 4(s-1) + 2 & \text{If } n \text{ is even} \\ 4s & \text{If } n \text{ is odd,} \end{cases} \tag{23}$$

*where two of the connected components of* G *are an isomorphic path graph.*

#### **Proof.**

1. It follows from Theorem 2 that *k*(G) = 2. The connected components of G are C<sup>1</sup> = (*V*1, *E*1) and C<sup>2</sup> = (*V*2, *E*2), as in the proof of Theorem 2 (1). It is then sufficient to show that C<sup>1</sup> ∼= C2. Using the function *<sup>f</sup>* defined in Lemma 5, we have for *<sup>a</sup><sup>i</sup> bj* , *akbl* ∈ *V*<sup>1</sup> that

$$\{a^i b^j, a^k b^l\} \in E\_1 \iff \{f(a^i b^j), f(a^k b^l)\} \in E\_{2\prime} \tag{24}$$

which leads to C<sup>1</sup> ∼= C2.

	- (a) Suppose *t* is an even number. The condition *t* > *r* implies that G has isolated points, and then, using Theorem 2, we have that G has at least two connected components. Let C<sup>1</sup> = (*V*1, *E*1) and C<sup>2</sup> = (*V*2, *E*2) for the connected components constructed in the proof of Theorem 2 (1). We prove first that |*V*1| = |*V*2|. In fact, we have that |*j* − *l*| = *t* or |*j* − *l*| = *t* − 1,

which implies that

$$j \in \{t - 1, \ldots, n - 1\} \cup \{0, \ldots, n - t\} =: A,\tag{25}$$

since *l* ∈ {0, . . . , *n* − 1}. It is clear that {*t* − 1, . . . , *n* − 1}∩{0, . . . , *n* − *t*} = ∅, therefore

$$|A| = 2(n - t) + 2.\tag{26}$$

On the other hand, it follows immediately that *j* ∈ *A* and *i* + *j* are even numbers if and only if *a<sup>i</sup> b<sup>j</sup>* ∈ *V*1, and then

$$|V\_1| = |A| = 2(n - t) + 2.\tag{27}$$

Analogously, |*V*2| = |*A*|, and we have |*V*1| = |*V*2|.

To demonstrate that C<sup>1</sup> ∼= C2, we consider again the function *<sup>f</sup>* defined in Lemma 5. Note that *f*(*V*1) = *V*2, and, since *f* is an isometry, we have the statement.

Finally, using Lemma 4, we have that |*E*| = 4(*n* − *t*) + 2, and the isomorphy between C<sup>1</sup> and C<sup>2</sup> implies that |*E*1| = |*E*2|. Further, note that the minimum value for |*E*1| and |*E*2| is 2(*n* − *t*) + 1. This proves that C<sup>1</sup> and C<sup>2</sup> are the unique connected components of G, which are not isolated points, and these are actually isomorphic path graphs.

The number of isolated points of G is |*Dn*|−|*V*1|−|*V*2| = 2*n* − 4(*n* − *t*) − 4 = −2*n* + 4*t* − 4, and, consequently, *k*(G) = −2*n* + 4*t* − 2 = −2*n* + 4*r* + 4*s* − 2. That is,


$$\{\{a^ib^j, a^kb^l\} \mid i+j \equiv 0 \bmod 2, k+l \equiv 1 \bmod 2 \land |i-k|+|j-l|=t\}, \tag{28}$$

is a subset of *E*. Let *V* be the set consisting of the non-isolated points of G. Using the same argument as in (a), we obtain

$$|V'| = 2|A| = 4(n - t) + 4.\tag{29}$$

By Lemma 4, we have that |*E*| = 4(*n* − *t*) + 2 , then, comparing |*V* | and |*E*| excluding the isolated points, it follows that G cannot be connected. Let *m* be an even number such that

$$\begin{cases} 0 \le m \le n - t - 1 & \text{if } n - t - 1 \text{ is even,} \\ 0 \le m \le n - t & \text{if } n - t - 1 \text{ is odd,} \end{cases} \tag{30}$$

and consider the subgraph C<sup>1</sup> = (*V*1, *E*1) of G with the following edges:

$$\{ab^{t+m-1},b^m\}, \{b^m,b^{t+m}\}, \{b^{t+m},ab^{m+1}\}, \{ab^{m+1},ab^{t+m+1}\}.$$

Then, C<sup>1</sup> is a connected component of G, and, furthermore,

$$|V\_1| = 2(n - t) + 2 \land |E\_1| = 2(n - t) + 1,\tag{31}$$

whence it is concluded that C<sup>1</sup> is a path graph.

As before, using the function *f* from Lemma 5, we have that there exists another connected component C<sup>2</sup> = (*f*(*V*1), *E*2), isomorphic to C1. Thus,

$$|E\_1| + |E\_2| = |E| \land |V\_1| + |V\_2| = |V'|. \tag{32}$$

This means that C<sup>1</sup> and C<sup>2</sup> are the unique connected components of G, and, analogously to the previous case, we have the same values for *k*(G).

The following corollary is a generalization of Theorem 3.

**Corollary 1.** *Let G be a two-generator group in the form* (10) *with n*, *m* ≥ 2*, and t be an odd number. Let further r be defined as in Theorem 2. If t* ≤ *r, then* G = (*G*, *E*) *is a bipartite graph.*

**Proof.** From Theorem 4, we have that G is connected. Now, we define the sets *V*<sup>1</sup> and *V*<sup>2</sup> as follows

$$V\_1 := \{ a^i b^j \mid i + j \equiv 0 \bmod 2 \}\tag{33}$$

$$V\_2 := \{ a^i b^j \mid i+j \equiv 1 \bmod 2 \}\tag{34}$$

It is immediate to verify that *V*<sup>1</sup> and *V*<sup>2</sup> form a bipartition of *G*, and G is a bipartite graph.

An illustration of the previous Corollary is presented in Figure 3.

**Figure 3.** The 3-graph of *G* = Z<sup>3</sup> × Z4.

**Corollary 2.** *Let n be an odd number, n* ≥ 5 *and t* = *<sup>n</sup>*+<sup>1</sup> <sup>2</sup> *.*


**Proof.** These statements follow directly from Theorem 4. Note that *t* = *r*.

1. From Lemma 4, it follows that

$$|E| = 4(n - \left(\frac{n+1}{2}\right)) + 2 = 2n = |D\_n|.\tag{35}$$

and then G is a cycle of even length.

2. G has two isomorphic connected components, say C<sup>1</sup> = (*V*1, *E*1) and C<sup>2</sup> = (*V*2, *E*2). Lemma 4 implies that

$$|E| = 4(n - \left(\frac{n+1}{2}\right)) + 2 = 2n,\tag{36}$$

and it follows that |*E*1| = |*E*2| = *n*, so G is constituted by two isomorphic cycles. Finally, note that each component has an odd number of vertices. Then, *χ*(G) = 3.

**Corollary 3.** *Let n be an even number, n* ≥ 2 *and t* = *<sup>n</sup>* <sup>2</sup> + 1*. Then, the t-graph of Dn consists of two isomorphic paths graphs.*

**Proof.** Using Theorem 4, and since *n* is an even number, we have that *r* = *<sup>n</sup>* <sup>2</sup> , and then *t* = *r* + 1, and *k*(G) = 2. The rest is clear.

**Corollary 4.** *Let n* ≥ 2 *and r be as in Theorem 4. Then, the t-graph of Dn is 2-chromatic if t* ≤ *r and t is an odd number or t* > *r.*

**Proof.** It follows immediately from Theorem 4 and Corollary 1.

**Corollary 5.** *The n-graph of Dn has* 2(*n* − 1) *connected components, and two of these are path graphs with two vertices.*

**Proof.** Let G = (*Dn*, *E*) be the *n*-graph of *Dn*. From Lemma 4, it follows that |*E*| = 2. Note that

$$\{a, b^{n-1}\}, \{ab^{n-1}, 1\} \in E. \tag{37}$$

The other 2*n* − 4 elements of *Dn* are isolated points, and the proof is complete.

#### **4. Some Questions and Conjectures**

Some open questions and conjectures are presented below.

**Question 1.** *Is it possible to characterize the t-graphs on two-generator groups, when t* > *r and r is as in Theorem 2?*

**Question 2.** *Is it possible to generalize a version of Theorem 2 for an n-generator group for n, an arbitrary natural number?*

**Question 3.** *It is possible to determine in a finite group the existence (or not) of a generating system with the conditions stated for the definition of the t-graphs?*

**Conjecture 1.** *With respect to Theorem 2, if m is an even number and t* ≤ *r, it follows that the two connected components of the t-graph* G *are isomorphic.*

**Conjecture 2.** *Let n* ≥ 2 *and r be as in Theorem 4. Then, the t-graph of Dn is 3-chromatic, if t* ≤ *r and t is an even number.*

**Conjecture 3.** *If G* = Z*<sup>n</sup>* × Z2*, then K*(*G*) = 2(2*t* − *n*) − 2*.*

**Conjecture 4.** *If G* = Z*<sup>n</sup>* × Z3*, then K*(*G*) = 3(2*t* − *n* − 2) − 2*.*

#### **5. Discussion**

In the present research, we introduce and investigate the *t*-graph on a finitely generated group *G*. It leads to an interesting combinatorial problem. We establish conditions for *t* to guarantee the existence of isolated points in the *t*-graph when *G* is a two-generator group. We also propose an expression to determine the number of the connected components of the *t*-graph. Other results have to do with the conditions that must be fulfilled for the *t*-graphs of the dihedral groups to be a path graph or a cycle. Consequently, we can characterize the chromatic number of the *t*-graph depending exclusively on the parity of *t*.

**Author Contributions:** Conceptualization, I.G.; methodology, I.G., G.D.-P. and A.T.-G.; software, A.T.-G.; validation, I.G., G.D.-P. and A.T.-G.; formal analysis, I.G. and A.T.-G. investigation, I.G., G.D.-P. and A.T.-G.; resources, I.G., G.D.-P. and A.T.-G.; writing—original draft preparation, I.G., G.D.-P. and A.T.-G.; writing—review and editing, I.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Obaidullah Wardak 1, Ayushi Dhama <sup>2</sup> and Deepa Sinha 1,\***


**Abstract:** We define an addition signed Cayley graph on a unitary addition Cayley graph *Gn* represented by Σ<sup>∧</sup> *<sup>n</sup>* , and study several properties such as balancing, clusterability and sign compatibility of the addition signed Cayley graph Σ<sup>∧</sup> *<sup>n</sup>* . We also study the characterization of canonical consistency of Σ<sup>∧</sup> *<sup>n</sup>* , for some *n*.

**Keywords:** addition signed Cayley graph Σ<sup>∧</sup> *n*

**MSC:** 05C 22; 05C 75
