Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Abstract

The power graph P(Zn) of Zn for a finite cyclic group Zn is a simple undirected connected graph such that two distinct nodes x and y in Zn are adjacent in P(Zn) if and only if x≠y and xi=y or yi=x for some non-negative integer i. In this article, we find the Laplacian eigenvalues of P(Zn) and show that P(Zn) is Laplacian integral (integer algebraic connectivity) if and only if n is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019).

Keywords

• algebraic connectivity
• Laplacian matrix
• Laplacian integral
• power graphs
• integers modulo group
• Euler’s totient function