Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group (3,q,r) for q < r Primes
Abstract
:1. Introduction
2. Preliminaries
2.1. Group Action
2.2. Coset Diagrams
- For both and generator s, there is exactly one incoming and one outgoing s-arc at the vertex x;
- For both x in X and , the path labeled 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 , of x is represented by unlabeled arcs joining the vertices to for , and to . 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 , of x is represented by arcs oriented in an anticlockwise direction, joining the vertices to for , and to ;
- The action of satisfies ;
- The action of t is given by the reflection in a vertical line of symmetry.
3. Alternating Quotients of Where Is Prime
4. Adjacency Matrices for the Coset Diagrams of Where
- (1)
- Trace
- (2)
- Trace for all n
- (1)
- Consider the adjacency matrices of coset diagrams resulting from the attachment of , and using handles . Let denote the diagonal elements of .
- (2)
- Let be a diagram with an adjacency matrix . By appending an additional single copy of and to the remaining handles of , we obtain with its corresponding adjacency matrix denoted as .
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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r | Diagram | Vertices | Cycle Structure of |
---|---|---|---|
11 | A | 33 | |
B | 34 | ||
C | 50 | ||
13 | A | 39 | |
B | 40 | ||
C | 58 | ||
A | |||
B | |||
C | |||
A | |||
B | |||
C | |||
A | |||
B | |||
C |
Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|
117 | 0 | 62 | ||
251 | 0 | 132 | ||
318 | 0 | 161 | ||
452 | 0 | 231 |
Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|
137 | 0 | 72 | 149.7791 | |
295 | 0 | 154 | 324.8379 | |
374 | 0 | 189 | 409.2571 |
Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|
187 | 0 | 102 | 199.3871 | |
257 | 0 | 142 | 286.1425 | |
327 | 0 | 182 | 365.0140 |
Adjacency Matrix | Vertices | Determinant | Trace | Energy |
---|---|---|---|---|
277 | 0 | 112 | 214.6266 | |
347 | 0 | 158 | 299.3381 | |
417 | 0 | 192 | 378.7628 |
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Younas, S.; Kousar, S.; Albaity, M.; Mahmood, T.
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group
Younas S, Kousar S, Albaity M, Mahmood T.
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group
Younas, Sajida, Sajida Kousar, Majed Albaity, and Tahir Mahmood.
2023. "Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group