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The diameter of a graph is the maximum distance between pairs of vertices inthe graph. A pair of vertices whose distance is equal to its diameter arecalled diametrically opposite vertices. The collection of shortest pathsbetween diametrically opposite vertices are referred as diametral paths. Inthis work, we enumerate the number of diametral paths for Fibonacci cubes,Lucas cubes and Alternate Lucas cubes. We present bijective proofs that showthat these numbers are related to alternating permutations and are enumeratedby Euler numbers.

Euler numbers and diametral paths in Fibonacci cubes, Lucas cubes and Alternate Lucas cubes