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

Abstract: The \textbf{Co-Prime Order Graph} $\Theta (G)$ of a given finite group is a simple undirected graph whose vertex set is the group $G$ itself, and any two vertexes x,y in $\Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ or prime. In this paper, we find a precise formula to count the degree of a vertex in 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,Dihedral group, Laplacian spectrum.