# Co-prime order graphs of finite Abelian groups and dihedral groups

Volume 23, Issue 3, pp 196--202
Publication Date: November 01, 2020 Submission Date: August 19, 2020 Revision Date: September 23, 2020 Accteptance Date: September 28, 2020
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### Authors

Amit Sehgal - Department of Mathematics, Pt. NRS Govt. College, Rohtak (Haryana), India. Manjeet - Department of Mathematics, Pt. NRS Govt. College, Rohtak (Haryana), India. Dalip Singh - Department of Mathematics, Maharshi Dayanand University, Rohtak (Haryana), India.

### Abstract

The 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 derive a precise formula to count the vertex's degree in the co-prime order graph of a finite Abelian group or dihedral group.We also investigate the Laplacian spectrum of the co-prime order graph $\Theta (G)$ when G is a finite Abelian p-group, ${\mathbb{Z}_p}^t \times {\mathbb{Z}_q}^s$ or a dihedral group $D_{p^n}$.

### Share and Cite

##### ISRP Style

Amit Sehgal, Manjeet, Dalip Singh, Co-prime order graphs of finite Abelian groups and dihedral groups, Journal of Mathematics and Computer Science, 23 (2021), no. 3, 196--202

##### AMA Style

Sehgal Amit, Manjeet, Singh Dalip, Co-prime order graphs of finite Abelian groups and dihedral groups. J Math Comput SCI-JM. (2021); 23(3):196--202

##### Chicago/Turabian Style

Sehgal, Amit, Manjeet,, Singh, Dalip. "Co-prime order graphs of finite Abelian groups and dihedral groups." Journal of Mathematics and Computer Science, 23, no. 3 (2021): 196--202

### Keywords

• Co-prime order graph
• finite Abelian group
• dihedral group
• Laplacian spectrum

•  05C25
•  05C50

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