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The unitary Cayley graph X n has vertex set Z n = {0, 1,. .. , n − 1}. Vertices a, b are adjacent, if gcd(a − b, n) = 1. For X n the chromatic number, the clique number, the independence number, the diameter and the vertex connectivity are determined. We decide on the perfectness of X n and show that all nonzero eigenvalues of X n are integers dividing the… (More)

A graph is called integral, if all of its eigenvalues are integers. For an abelian group Γ we show that Cay(Γ, S) is integral, if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ. The converse is proven for cyclic groups. A finite group Γ is called Cayley integral, if every undirected Cayley graph over Γ is integral. We determine all… (More)

The energy of a graph was introduced by 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. These are Cayley graphs on cyclic groups (i.e. there adjacency matrix is circulant) each of whose eigenvalues is an integer. Given an arbitrary prime… (More)

An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Z m 1 ⊗ · · · ⊗ Z mr be an abelian group represented as the direct product of cyclic groups Z m i of order m i such that all greatest common divisors gcd(m i , m j) ≤ 2 for i = j. We prove that a Cayley graph Cay(Γ, S) over Γ is integral, if and only if S ⊆ Γ belongs to… (More)

Sudoku graphs have only 5 or 6 distinct eigenvalues and all of them are integers. Moreover, the associated eigenspaces admit bases with entries from the set {0, 1, −1}.

For every cograph there exist bases of the eigenspaces for the eigenvalues 0 and −1 that consist only of vectors with entries from {0, 1, −1}, a property also exhibited by other graph classes. Moreover, the multiplicities of the eigenvalues 0 and −1 of a cograph can be determined by counting certain vertices of the associated cotree.

- Markus Nick, Sören Schneickert, Jürgen Grotepaß, Helmut Hamfeld, Thomas Rose, Torsten Sander +3 others
- LWA
- 2006

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.