Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

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 t...

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Veröffentlicht in:Mathematics (Basel) Jg. 12; H. 14; S. 2175
1. Verfasser: Rather, Bilal Ahmad
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Basel MDPI AG 01.07.2024
Schlagworte:
Algebra
algebraic connectivity
Analysis
Combinatorial analysis
Connectivity
Eigenvalues
Euler’s totient function
Graph theory
Graphs
Group theory
Integers
integers modulo group
Laplacian integral
Laplacian matrix
power graphs
United Arab Emirates
ISSN:2227-7390, 2227-7390
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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).
AbstractList The power graph P(Z[sub.n] ) of Z[sub.n] for a finite cyclic group Z[sub.n] is a simple undirected connected graph such that two distinct nodes x and y in Z[sub.n] are adjacent in P(Z[sub.n] ) if and only if x≠y and x[sup.i] =y or y[sup.i] =x for some non-negative integer i. In this article, we find the Laplacian eigenvalues of P(Z[sub.n] ) and show that P(Z[sub.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).
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).
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 xi=y or yi=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).
Audience Academic
Author Rather, Bilal Ahmad
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10.1017/CBO9780511801518
10.1080/00029890.2009.11920976
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Snippet 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...
The power graph P(Z[sub.n] ) of Z[sub.n] for a finite cyclic group Z[sub.n] is a simple undirected connected graph such that two distinct nodes x and y in...
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...
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SubjectTerms Algebra
algebraic connectivity
Analysis
Combinatorial analysis
Connectivity
Eigenvalues
Euler’s totient function
Graph theory
Graphs
Group theory
Integers
integers modulo group
Laplacian integral
Laplacian matrix
power graphs
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