Geodesic
On Order Prime Divisor Graphs of Finite Groups
Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 2, pp. 419-437.

Voir la notice de l'article provenant de la source Library of Science

The order prime divisor graph 𝒫𝒟(G) of a finite group G is a simple graph whose vertex set is G and two vertices a, b ∈ G are adjacent if and only if either ab = e or o(ab) is some prime number, where e is the identity element of the group G and o(x) denotes the order of an element x ∈ G. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph 𝒫𝒟(G) of a group G. Concentrating on the graph 𝒫𝒟(D_n), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of 𝒫𝒟(ℤ_n), 𝒫𝒟(S_n), 𝒫𝒟(A_n).
Keywords: group, dihedral group, complete graph, Eulerian graph, regular graph, planar graph, order prime divisor graph 
@article{DMGAA_2021_41_2_a13,
     author = {Sen, Mridul K. and Maity, Sunil K. and Das, Sumanta},
     title = {On {Order} {Prime} {Divisor} {Graphs} of {Finite} {Groups}},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {419--437},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2021_41_2_a13/}
}
TY  - JOUR
AU  - Sen, Mridul K.
AU  - Maity, Sunil K.
AU  - Das, Sumanta
TI  - On Order Prime Divisor Graphs of Finite Groups
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2021
SP  - 419
EP  - 437
VL  - 41
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2021_41_2_a13/
LA  - en
ID  - DMGAA_2021_41_2_a13
ER  - 
%0 Journal Article
%A Sen, Mridul K.
%A Maity, Sunil K.
%A Das, Sumanta
%T On Order Prime Divisor Graphs of Finite Groups
%J Discussiones Mathematicae. General Algebra and Applications
%D 2021
%P 419-437
%V 41
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2021_41_2_a13/
%G en
%F DMGAA_2021_41_2_a13
Sen, Mridul K.; Maity, Sunil K.; Das, Sumanta. On Order Prime Divisor Graphs of Finite Groups. Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 2, pp. 419-437. http://geodesic.mathdoc.fr/item/DMGAA_2021_41_2_a13/

[1] J. Bosák, The Graphs of Semigroups, in: Theory of Graphs and Application (Academic Press, New York, 1964) 119–125.

[2] B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969) 241–247. https://doi.org/10.21136/CMJ.1969.100891

[3] I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009) 5381–5392. https://doi.org/10.1016/j.disc.2008.11.034

[4] I. Chakrabarty, S. Ghosh and M.K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (2009) 410–426. https://doi.org/10.1007/s00233-008-9132-y

[5] D.S. Dummit and R.M. Foote, Abstract Algebra, Third Edition (John Wiley and Sons, Inc., New York, 2004).

[6] A.V. Kelarev and S.J. Quinn, A combinatorial property and power graphs of groups, Contributions to General Algebra 12 (2000) 229–235.

[7] M. Sattanathan and R. Kala, An introduction to order prime graph, Int. J. Contemp. Math. Sciences 4 (2009) 467–474. https://doi.org/10.1111/j.1439-0507.1967.tb02798.x

[8] D.B. West, Introduction to Graph Theory, Second Edition (Pearson India Education Services Pvt. Ltd., 2017).

[9] B. Zelinka, Intersection graphs of finite Abelian groups, Czech. Math. J. 25 (1975) 171–174. https://doi.org/10.21136/CMJ.1975.101307