Characterizations of algebraic and vertex connectivity of power graph of finite cyclic groups

Document Type : Research Paper

Authors

Department of Mathematics, University of Mumbai, Mumbai, India

Abstract

The Power graph of a group $G$ is a graph $\mathcal{P}(G)$ with vertex set $G$ and two vertices $x$ and $y$, $x \neq y$ are adjacent if there exists some integer $k$ such that $x=y^k$ or $y=x^k$. We denote the vertex connectivity of power graph $\mathcal{P}(G)$ by $\mathcal{K}(\mathcal{P}(G))$ and the algebraic connectivity of power graph $\mathcal{P}(G)$ by $\lambda_{n-1}(\mathcal{P}(G))$. This paper investigates the upper bound for the vertex connectivity and the algebraic connectivity of $\mathcal{P}(\mathbb{Z}_{n})$. Moreover, we discuss the equivalent conditions for $\mathcal{P}(\mathbb{Z}_{n})$ to be Laplacian integral. Further the conjecture for an upper bound of the algebraic connectivity of $\mathcal{P}(\mathbb{Z}_{n})$ is posed in this article.

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References

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