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

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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 
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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/