Computing with graphs and groups

The Density of Sets Avoiding Distance 1 in Euclidean Space

“…A direct computation of α(J(n, w, i)) for all w, i turns successful only up to n = 10 (we have performed the computations using the package GRAPE of the computational system GAP, that deals with graphs with symmetries [28]; indeed, the graphs J(n, w, i) are invariant under the group of permutations of the n coordinates). On the other hand, for the graphs J(n, 3, 1), there is an explicit formula due to Erdős and Sós (see [14,Lemma 18]), but the number of vertices in this case, which is roughly equal to n 3 , it too small to lead to a good bound.…”

Constructing Embeddings and Isomorphisms of Finite Abstract Semigroups

“…To our best knowledge, the SubSemi package is the first software tool capable of calculating general semigroup embeddings. For the isomorphism calculation, we can compare our method to SmallestMultiplicationTable function in the Semigroups package [25] (based on the SmallestImageSet function in Gap [15] and depending on the GRAPE package [27]). This function calculates the smallest multiplication table in lexicographic ordering for the semigroup (by calculating a group action orbit of the symmetric group), which can be used for isomorphism testing.…”

On locally primitive Cayley graphs of finite simple groups

“…For group pairs in (b), computation by using GAP and the package GRAPE [14] shows that there exists no such graph.…”