The energy of unitary cayley graphs

✍ Scribed by Aleksandar Ilić

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 146 KB
• Volume: 431
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.06.025

✦ Synopsis

where E(G) denotes the energy of G. The unitary Cayley graph X n has vertex set Z n = {0, 1, 2, . . . , n -1} and vertices a and b are adjacent, if gcd(ab, n) = 1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph X n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that X n is hyperenergetic if and only if n has at least two distinct prime factors and n / = 2p, where p is a prime number. By extending this approach, for every fixed k ∈ N, we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.