The energy of integral circulant graphs with prime power order

Abstract

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a, b} : a, b ∈ Zn, gcd(a − b, n) ∈ D}. For an integral circulant graph on p s vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed p s and varying divisor sets D.2010 Mathematics Subject Classification. Primary. 05C50, Secondary. 15A18.

The maximal energy of classes of integral circulant graphs

“…A straightforward consequence of (1) is that E min (p s ) is attained precisely for the singleton divisor sets D = {p t } with 0 ≤ t ≤ s − 1 (cf. [20], Theorem 3.1).…”

“…Over the years, the general structural properties of integral circulant graphs have been well researched [11], [8], [24], [16], [2], [3], [17], [12], [7], [4]. Due to the connection with quantum physics, emphasis has lately been placed on researching the energy of integral circulant graphs [23], [15], [19], [5], [18], [20], [21].…”

Distance powers of unitary Cayley graphs

“…Until now, the energies of graphs have been studied extensively. We refer the reader to [4,[11][12][13][14][15][16]19,20,[24][25][26][27]29,30,33,34] .…”

Integral circulant graphs of prime power order with maximal energy

“…In [18] the authors proved an explicit formula for the energy of ICG(p s , D) for any prime power p s and any divisor set D = {p a 1 , p a 2 , . .…”

“…Not surprisingly, the class of graphs ICG(p s , D) contains also non-hyperenergetic elements, termed hypoenergetic. For the minimal energy E min (n) of all integral circulant graphs with n vertices it has been shown in [18] that…”