EN
Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex
Abstract
This paper investigates a particular class of digraph groups that are defined by non-empty balanced presentations. Each relation is expressed in the form R(x,y), where x and y are distinct generators, and R(⋅,⋅) is based on a fixed cyclically reduced word R(a,b) involving both a and b. A directed graph is constructed for each such presentation, where vertices correspond to generators and edges represent the relations. In previous research, Cihan identified 35 families of digraphs that satisfy |V(Γ)|=|A(Γ)|-1, of which 11 of them do not contain leaves. This paper demonstrates that, with two exceptions, the rank of the associated groups is either 1 or 2.
Keywords
References
- [1] Cuno J., Williams G., A class of digraph groups defined by balanced presentations, Journal of Pure and Applied Algebra., 224(8) (2020) 106342.
- [2] Cihan M.S., Williams G., Finite groups defined by presentations in which each defining relator involves exactly two generators, Journal of Pure and Applied Algebra 228 (4) (2024) 107499.
- [3] Johnson D.L., Topics in the theory of group presentations, London Mathematical Society Lecture Note Series, 42. Cambridge University Press, (1980).
- [4] Johnson D.L., Robertson E.F., Finite groups of deficiency zero, In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, (36) 1979 275-289.
- [5] Cihan M.S., Digraph groups corresponding to digraphs with one more vertex than arcs, European Journal of Science and Technology., (41) (2022) 31–35.
- [6] Pride S.J., Groups with presentations in which each defining relator involves exactly two generators, J. Lond. Math. Soc., II. Ser. 36 (1-2) (1987) 245–256.
- [7] Bogley W.A., Williams G., Efficient finite groups arising in the study of relative asphericity, Math. Z. 284(1) (2016) 507–535.
- [8] Cihan M.S., Digraph Groups and Related Groups, Doctoral dissertation, University of Essex, 2022.
Details
Primary Language
English
Subjects
Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)
Journal Section
Research Article
Authors
Publication Date
June 30, 2025
Submission Date
March 12, 2025
Acceptance Date
June 11, 2025
Published in Issue
Year 2025 Volume: 46 Number: 2
APA
Cihan, M. S. (2025). Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex. Cumhuriyet Science Journal, 46(2), 410-423. https://doi.org/10.17776/csj.1656241
AMA
1.Cihan MS. Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex. CSJ. 2025;46(2):410-423. doi:10.17776/csj.1656241
Chicago
Cihan, Mehmet Sefa. 2025. “Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex”. Cumhuriyet Science Journal 46 (2): 410-23. https://doi.org/10.17776/csj.1656241.
EndNote
Cihan MS (June 1, 2025) Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex. Cumhuriyet Science Journal 46 2 410–423.
IEEE
[1]M. S. Cihan, “Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex”, CSJ, vol. 46, no. 2, pp. 410–423, June 2025, doi: 10.17776/csj.1656241.
ISNAD
Cihan, Mehmet Sefa. “Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex”. Cumhuriyet Science Journal 46/2 (June 1, 2025): 410-423. https://doi.org/10.17776/csj.1656241.
JAMA
1.Cihan MS. Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex. CSJ. 2025;46:410–423.
MLA
Cihan, Mehmet Sefa. “Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex”. Cumhuriyet Science Journal, vol. 46, no. 2, June 2025, pp. 410-23, doi:10.17776/csj.1656241.
Vancouver
1.Mehmet Sefa Cihan. Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex. CSJ. 2025 Jun. 1;46(2):410-23. doi:10.17776/csj.1656241