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

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

2011-09-16

arxiv.org/abs/1109.3616v1

Mathematics

Combinatorics

Scientific paper

The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values 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 $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z},\, \gcd(a-b,n)\in {\cal D}\}$. Given an arbitrary prime power $p^s$, we determine all divisor sets maximising the energy of an integral circulant graph of order $p^s$. This enables us to compute the maximal energy $\Emax{p^s}$ among all integral circulant graphs of order $p^s$.

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