#
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group ${\u25b5}^{*}$ (3,q,r) for q < r Primes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Group Action

#### 2.2. Coset Diagrams

- For both $x\in X$ and generator s, there is exactly one incoming and one outgoing s-arc at the vertex x;
- For both x in X and $r={s}_{{i}_{1}}\dots {s}_{{i}_{k}}\in R$, the path labeled ${s}_{{i}_{1}},\dots ,{s}_{{i}_{k}}$ starting at x is closed.

- q-cycles of y are depicted by shaded q-gons whose vertices are permuted in an anticlockwise direction, and fixed points of y by heavy dots;
- A p-cycle, say $({\alpha}_{1},\dots ,{\alpha}_{p})$, of x is represented by unlabeled arcs joining the vertices ${\alpha}_{i}$ to ${\alpha}_{i+1}$ for $i=1\dots (p-1)$, and ${\alpha}_{p}$ to ${\alpha}_{1}$. The loops representing fixed points of x are omitted from the diagram.

- The q-cycles of y are represented by shaded q-gons permuted in an anticlockwise direction, and fixed points of y by heavy dots;
- A p-cycle, say $({\alpha}_{1},\dots ,{\alpha}_{p})$, of x is represented by arcs oriented in an anticlockwise direction, joining the vertices ${\alpha}_{i}$ to ${\alpha}_{i+1}$ for $i=1\dots (p-1)$, and ${\alpha}_{p}$ to ${\alpha}_{1}$;
- The action of $xy$ satisfies ${\left(xy\right)}^{r}=1$;
- The action of t is given by the reflection in a vertical line of symmetry.

**Proposition**

**1.**

## 3. Alternating Quotients of ${\u25b5}^{*}(\mathbf{3},\mathbf{7},\mathbf{r})$ Where $\mathbf{r}\ge \mathbf{11}$ Is Prime

**Theorem**

**1.**

**Lemma**

**1.**

## 4. Adjacency Matrices for the Coset Diagrams of ${\u25b5}^{*}(\mathbf{3},\mathbf{7},\mathbf{r})$ Where $\mathbf{r}\ge \mathbf{11}$

**Theorem**

**2.**

- (1)
- Trace$\left({M}_{{D}_{2n}}\right)=Trace\left({M}_{{D}_{2n-1}}\right)+70$
- (2)
- Trace$\left({M}_{{D}_{2n+1}}\right)=Trace\left({M}_{{D}_{2n}}\right)+29$ for all n

**Proof**

**.**

- (1)
- Consider the adjacency matrices ${M}_{{D}_{2n-1}}$ of coset diagrams ${\mathfrak{D}}_{\mathfrak{2}\mathfrak{n}-\mathfrak{1}}$ resulting from the attachment of $A\left(11\right)$, $B\left(11\right)$ and $C\left(11\right)$ using handles ${H}_{1}$. Let ${\stackrel{\xb4}{d}}_{1},{\stackrel{\xb4}{d}}_{2},\dots ,{\stackrel{\xb4}{d}}_{2n-1}$ denote the diagonal elements of ${M}_{{D}_{2n-1}}$.

- (2)
- Let ${\mathfrak{D}}_{\mathfrak{2}\mathfrak{n}}$ be a diagram with an adjacency matrix ${M}_{{D}_{2n}}$. By appending an additional single copy of $A\left(11\right)$ and $B\left(11\right)$ to the remaining handles ${H}_{1}$ of ${\mathfrak{D}}_{\mathfrak{2}\mathfrak{n}}$, we obtain ${\mathfrak{D}}_{\mathfrak{2}\mathfrak{n}+\mathfrak{1}}$ with its corresponding adjacency matrix denoted as ${M}_{{D}_{2n+1}}$.

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Jones, G.A.; Singerman, D. Complex Functions an Algebraic and Geometric Viewpoint; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Penafort, G.S. Reflection maps. Math. Ann.
**2020**, 378, 559–598. [Google Scholar] [CrossRef] - Miller, G.A. On the groups generated by two operators. Bull. Amer. Math. Soc.
**1901**, 7, 424–426. [Google Scholar] [CrossRef] - Conder, M.D.E. More on generators for alternating and symmetric groups. Quart. J. Math.
**1981**, 32, 137–163. [Google Scholar] [CrossRef] - Mushtaq, Q.; Rota, G.C. Alternating groups as quotients of two generator groups. Adv. Math.
**1992**, 96, 113–121. [Google Scholar] [CrossRef] - Mushtaq, Q.; Servatius, H. Permutation representations of the symmetry groups of regular hyperbolic tessellations. J. Lond. Math. Soc.
**1993**, 2, 77–86. [Google Scholar] [CrossRef] - Everitt, B. Permutation representations of the (2,4,r) triangle groups. Bull. Austral. Math. Soc.
**1994**, 49, 499–511. [Google Scholar] [CrossRef] - Everitt, B. Alternating quotients of Fuchsian groups. J. Algebra
**2000**, 223, 457–476. [Google Scholar] [CrossRef] - Walter, C.D. Adjacency matrices. SIAM J. Algebr. Discret. Methods.
**1986**, 7, 18–29. [Google Scholar] [CrossRef] - Patne, R.M.; Avachar, G.R. Adjacency matrices of generalized composition and generalized disjunction of graphs. Adv. Math. Sci. J.
**2020**, 9, 1281–1291. [Google Scholar] [CrossRef] - Harary, F. The determinant of the adjacency matrix of a graph. SIAM Rev.
**1962**, 4, 202–210. [Google Scholar] [CrossRef] - Elspas, B.; Turner, J. Graphs with circulant adjacency matrices. J. Combin. Theory
**1970**, 9, 297–307. [Google Scholar] [CrossRef] - Kaveh, A.; Rahami, H. Block diagonalization of adjacency and Laplacian matrices for graph product, application in structrual mechanics. Internat. J. Numer. Methods Engrg.
**2006**, 68, 33–63. [Google Scholar] [CrossRef] - Gutman, I.; Zhou, B. Laplacian energy of a graph. Linear Algebra Appl.
**2006**, 414, 29–37. [Google Scholar] [CrossRef] - Liu, J.; Li, X. Hermitian-adjacency matrices and Hermitian energies of mixed graphs. Linear Algebra Appl.
**2015**, 466, 182–207. [Google Scholar] [CrossRef] - Das, K.C.; Mojalal, S.A. On energy and Laplacian energy of graphs. Electron. J. Linear Algebra
**2016**, 31, 167–186. [Google Scholar] [CrossRef] - Khan, M.; Umar, S.; Broumi, S. Laplacian energy of a complex neutrosophic graph. Stud. Fuzzy Soft Comput.
**2019**, 369, 203–232. [Google Scholar] - Cokilavany, D. Extended energy of some special class of unicyclic graphs. J. Adv. Res. Dyn. Control Syst.
**2020**, 12, 1808–1815. [Google Scholar] [CrossRef] - Ramezani, F.; Stanić, Z. Some upper bounds for the net Laplacian index of a signed graph. Bull. Iran. Math. Soc.
**2022**, in press. [Google Scholar] [CrossRef] - Huang, N.; Xiao, L.; Xu, Y. Spectral analysis of the adjacency matrix of random geometric graphs. Phys. Life Rev.
**2019**, 31, 240–256. [Google Scholar] - Litvak, A.E.; Lytova, A.; Tikhomirov, K.; Tomczak-Jaegermann, N.; Youssef, P. Structure of eigenvectors of random regular digraphs. Trans. Amer. Math. Soc.
**2019**, 37, 8097–8172. [Google Scholar] [CrossRef] - Mushtaq, Q.; Rafiq, A. Adjacency matrices of PSL(2, 5) and resemblance of its coset diagrams with Fullerene C
_{60}. Chin. Acad. Sci.**2013**, 20, 541–552. [Google Scholar] [CrossRef] - Rafiq, A.; Mushtaq, Q. Coset diagrams of the modular group and continued fractions. Compt. Rend. Math.
**2019**, 357, 655–663. [Google Scholar] [CrossRef] - Mushtaq, Q.; Razaq, A.; Yousaf, A. On contraction of vertices of the circuits in coset diagrams for PSL(2,Z). Proc. Indian Acad. Sci. Math. Sci.
**2019**, 129, 13. [Google Scholar] [CrossRef] - Rose, J.S. A Course on Group Theory; Cambridge University Press: Cambridge, UK, 1978. [Google Scholar]
- Wolf, T.R. Group actions and non-vanishing elements in solvable groups. J. Group Theory
**2020**, 23, 1103–1109. [Google Scholar] [CrossRef] - Izquierdo, M.; Reyes-Carocca, S.; Rojas, A.M. On families of Riemann surfaces with automorphisms. J. Pure Appl. Algebra
**2021**, 225, 106704. [Google Scholar] [CrossRef] - Ashiq, M.; Mushtaq, Q. Coset diagrams for a homomorphic image of Δ(3,3,k). Acta Math. Sci. Ser. B Engl. Ed.
**2008**, 28, 363–370. [Google Scholar] [CrossRef] - White, A.T. Graphs, Groups and Surfaces; Elsevier: Amsterdam, The Netherlands, 1984. [Google Scholar]
- Wielandt, H. Finite Permutation Groups; Academic Press: San Diego, CA, USA, 1964. [Google Scholar]

r | Diagram | Vertices | Cycle Structure of ${\mathit{x}}^{-1}\mathit{y}$ |
---|---|---|---|

11 | A | 33 | $\left({a}_{1}{a}_{2}9\right)\left({{a}_{1}}^{\prime}{{a}_{2}}^{\prime}9\right)\phantom{\rule{0.277778em}{0ex}}11$ |

B | 34 | $\left({b}_{1}{b}_{2}9\right)\left({{b}_{1}}^{\prime}{{b}_{2}}^{\prime}9\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}10$ | |

C | 50 | $\left({c}_{1}{c}_{2}7\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{5}^{2}\phantom{\rule{0.277778em}{0ex}}6\phantom{\rule{0.277778em}{0ex}}23$ | |

13 | A | 39 | $\left({a}_{1}{a}_{2}11\right)\left({{a}_{1}}^{\prime}{{a}_{2}}^{\prime}11\right)\phantom{\rule{0.277778em}{0ex}}13$ |

B | 40 | $\left({b}_{1}{b}_{2}11\right)\left({{b}_{1}}^{\prime}{{b}_{2}}^{\prime}11\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}12$ | |

C | 58 | $\left({c}_{1}{c}_{2}9\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{7}^{2}\phantom{\rule{0.277778em}{0ex}}8\phantom{\rule{0.277778em}{0ex}}23$ | |

$11+6h$ | A | $33+21h$ | $\left({a}_{1}{a}_{2}10\right)\left({{a}_{1}}^{\prime}{{a}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}{6}^{3}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}12$ |

B | $34+21h$ | $\left({b}_{1}{b}_{2}10\right)\left({{b}_{1}}^{\prime}{{b}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{6}^{3}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}11$ | |

C | $50+28h$ | $\left({c}_{1}{c}_{2}8\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{6}^{6}{7}^{1+4(h-1)}\mathbf{23}$ | |

$13+6h$ | A | $39+21h$ | $\left({a}_{1}{a}_{2}10\right)\left({{a}_{1}}^{\prime}{{a}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}{8}^{3}\phantom{\rule{0.277778em}{0ex}}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}12$ |

B | $40+21h$ | $\left({b}_{1}{b}_{2}10\right)\left({{b}_{1}}^{\prime}{{b}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{8}^{3}\phantom{\rule{0.277778em}{0ex}}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}11$ | |

C | $58+28h$ | $\left({c}_{1}{c}_{2}8\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{6}^{2}\phantom{\rule{0.277778em}{0ex}}{7}^{1+4(h-1)}\phantom{\rule{0.277778em}{0ex}}{8}^{4}\phantom{\rule{0.277778em}{0ex}}\mathbf{23}$ | |

$15+6h$ | A | $41+21h$ | $\left({a}_{1}{a}_{2}10\right)\left({{a}_{1}}^{\prime}{{a}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}{10}^{3}\phantom{\rule{0.277778em}{0ex}}12$ |

B | $42+21h$ | $\left({b}_{1}{b}_{2}10\right)\left({{b}_{1}}^{\prime}{{b}_{2}}^{\prime}10\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{7}^{3(h-1)}\phantom{\rule{0.277778em}{0ex}}{10}^{3}\phantom{\rule{0.277778em}{0ex}}11$ | |

C | $60+21h$ | $\left({c}_{1}{c}_{2}8\right)\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.277778em}{0ex}}{6}^{2}\phantom{\rule{0.277778em}{0ex}}{7}^{1+4(h-1)}\phantom{\rule{0.277778em}{0ex}}{10}^{4}\phantom{\rule{0.277778em}{0ex}}\mathbf{23}$ |

Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|

${M}_{{\mathfrak{D}}_{1}}$ | 117 | 0 | 62 | $127.7403$ |

${M}_{{\mathfrak{D}}_{2}}$ | 251 | 0 | 132 | $272.4878$ |

${M}_{{\mathfrak{D}}_{3}}$ | 318 | 0 | 161 | $341.9508$ |

${M}_{{\mathfrak{D}}_{4}}$ | 452 | 0 | 231 | $493.3243$ |

Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|

${M}_{{\left({\mathfrak{D}}_{\mathfrak{1}}BP\right)}_{0}}$ | 137 | 0 | 72 | 149.7791 |

${M}_{{\left({\mathfrak{D}}_{\mathfrak{2}}BP\right)}_{2}}$ | 295 | 0 | 154 | 324.8379 |

${M}_{{\left({\mathfrak{D}}_{\mathfrak{3}}BP\right)}_{3}}$ | 374 | 0 | 189 | 409.2571 |

Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|

${M}_{{\left({\mathfrak{D}}_{B1H}\right)}_{0}}$ | 187 | 0 | 102 | 199.3871 |

${M}_{{\left({\mathfrak{D}}_{B2H}\right)}_{0}}$ | 257 | 0 | 142 | 286.1425 |

${M}_{{\left({\mathfrak{D}}_{B3H}\right)}_{0}}$ | 327 | 0 | 182 | 365.0140 |

Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|

${M}_{{\left({\mathfrak{D}}_{B2HP}\right)}_{0}}$ | 277 | 0 | 112 | 214.6266 |

${M}_{{\left({\mathfrak{D}}_{B3HP}\right)}_{0}}$ | 347 | 0 | 158 | 299.3381 |

${M}_{{\left({\mathfrak{D}}_{B4HP}\right)}_{0}}$ | 417 | 0 | 192 | 378.7628 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Younas, S.; Kousar, S.; Albaity, M.; Mahmood, T.
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group *q*,*r*) for *q* < *r* Primes. *Axioms* **2023**, *12*, 1128.
https://doi.org/10.3390/axioms12121128

**AMA Style**

Younas S, Kousar S, Albaity M, Mahmood T.
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group *q*,*r*) for *q* < *r* Primes. *Axioms*. 2023; 12(12):1128.
https://doi.org/10.3390/axioms12121128

**Chicago/Turabian Style**

Younas, Sajida, Sajida Kousar, Majed Albaity, and Tahir Mahmood.
2023. "Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group *q*,*r*) for *q* < *r* Primes" *Axioms* 12, no. 12: 1128.
https://doi.org/10.3390/axioms12121128