Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Bilal Ahmad Rather ()
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Bilal Ahmad Rather: Department of Mathematical Sciences, College of Science, United Arab Emirate University, Al Ain 15551, United Arab Emirates

Mathematics, 2024, vol. 12, issue 14, 1-12

Abstract: The power graph P ( Z n ) of Z n for a finite cyclic group Z n is a simple undirected connected graph such that two distinct nodes x and y in Z n are adjacent in P ( Z n ) if and only if x ≠ y and x i = y or y i = x for some non-negative integer i . In this article, we find the Laplacian eigenvalues of P ( Z n ) and show that P ( Z n ) 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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