Integral circulant graphs of prime power order with maximal energy

Details Integral circulant graphs of prime power order with maximal energy Integral circulant graphs of prime power order with maximal energy

2011-12-14

arxiv.org/abs/1112.3205v1

Linear Algebra Appl., Vol 435 (2011), pp. 3212-3232

Mathematics

Combinatorics

25 pages

Scientific paper

10.1016/j.laa.2011.05.039

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 Zn and edge set {{a, b} : a, b in Zn; gcd(a - b, n) in D}. Using tools from convex optimization, we study the maximal energy among all integral circulant graphs of prime power order ps and varying divisor sets D. Our main result states that this maximal energy approximately lies between s(p - 1)p^(s-1) and twice this value. We construct suitable divisor sets for which the energy lies in this interval. We also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets.

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