Order divisor graphs of finite groups

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we obtain the following results. The order divisor graph OD(G) of a finite group G is a star graph if and only if every non-identity element of G has prime order. For an abelian group G, OD(G) is a star graph if and only if G is elementary abelian . The order divisor graph of the unit group U(Zn) is a star graph if and only if n | 24 . The order divisor graph OD(Z n ) is a star graph if and only if n is prime. The order divisor graph OD(G) of a (finite) group G is a star graph if and only if G is a p-group of exponent p, or a non-nilpotent group of order p a q, or it is isomorphic to the simple group A 5 . The order divisor graph of the dihedral group D n (n ≥ 3) is a star graph S 2n if and only if n is prime . If G is a finite p-group of order p n then OD(G) is a complete multi-partite graph. If G is a finite cyclic group of order p n then OD(G) is complete (n+1)-partite graph . If G is a finite cyclic group of order p n , then χ(OD(G)) = n + 1. If G is a cyclic group of order p 1 p 2 , where p 1 ,p 2 are distinct primes, then OD(G) is a sequential join of graphs . Similarly, if G is a cyclic group of order p 1 p 2 p 3 , where p 1 ,p 2 ,p 3 are distinct primes, then OD(G) is obtained by certain type of sequential and cyclic joins.

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