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Year 2025, Volume: 46 Issue: 2, 410 - 423, 30.06.2025
https://doi.org/10.17776/csj.1656241

Abstract

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.

Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex

Year 2025, Volume: 46 Issue: 2, 410 - 423, 30.06.2025
https://doi.org/10.17776/csj.1656241

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.

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.
There are 8 citations in total.

Details

Primary Language English
Subjects Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)
Journal Section Natural Sciences
Authors

Mehmet Sefa Cihan 0000-0002-4303-9023

Publication Date June 30, 2025
Submission Date March 12, 2025
Acceptance Date June 11, 2025
Published in Issue Year 2025Volume: 46 Issue: 2

Cite

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