The maximal energy of classes of integral circulant graphs

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, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Z n and edge set {{a, b} : a, b ∈ Z n , gcd(a − b, n) ∈ D}. For a fixed prime power n = p s and a fixed divisor set size |D| = r, we analyze the maximal energy among all matching integral circulant graphs. Let p a 1 < p a 2 < . . . < p ar be the elements of D. It turns out that the differences d i = a i+1 − a i between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all d i equal q := s−1 r−1 , or at most the two differences [q] and [q + 1] may occur; (ii) there are rules governing the sequence d 1 , . . . , d r−1 of consecutive differences. For particular choices of s and r these conditions already guarantee maximal energy and its value can be computed explicitly.

The exact maximal energy of integral circulant graphs with prime power order

“…The following result is a corollary to Proposition 3.1 in [25]. Yet, in view of the new perspective of this exposition and for the convenience of the reader, we provide a short proof of it.…”

“…Proof. This is a special case of [25], Proposition 3.2. Instead of becoming acquainted with the notation there, the reader might be well advised to look at the involved delta tableaux and do some calculations similar to those above.…”

“…In [25] the authors used combinatorial instead of analytic arguments to refine the earlier approximative results on minimisers of h p,r for fixed values of r. This shed more light on the structure of these minimisers, which can be regarded as the result of a repeated balancing process. In several cases this process allowed us to obtain accurate results after only one or two balancing steps.…”

“…where a (m) is one of the maximising exponent tuples to be found in Theorem 1.1. Let us point out that, contrary to earlier strategies applied in [24] and [25], we now compare the energies related to admissible exponent tuples of d i f f e r e n t lengths. We recommend to take a look at the illustrative Example 3.1.…”

The asymptotic trace norm of random circulants and the graph energy

“…Moreover, we will produce explicit estimates on the rate of convergence and on the probabilities of large deviations. We chose to focus the exposition on circulants rather than Töeplitz matrices not only because the estimates are cleaner, but also because circulant graphs are easier to visualize, and their energies were a subject of much research lately [4,13,24,37].…”