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The \textbf{Co-Prime Order Graph} $\Theta (G)$ of a given finite group is asimple undirected graph whose vertex set is the group $G$ itself, and any twovertexes x,y in $\Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ orprime. In this paper, we find a precise formula to count the degree of a vertexin the Co-Prime Order graph of a finite abelian group or Dihedral group$D_n$.We also investigate the Laplacian spectrum of the Co-Prime Order Graph$\Theta (G)$ when G is finite abelian p-group, ${\mathbb{Z}_p}^t \times{\mathbb{Z}_q}^s$ or Dihedral group $D_{p^n}$.  Key Words and Phrases: Co-Prime Order graph,finite abelian group,Dihedralgroup, Laplacian spectrum.

Co-Prime Order graph of a finite abelian Group and Dihedral Group