N°318 – Electroencephalographic changes in children with autism spectrum disorders

For integer k≥2 and prime power q, the algebraic bipartite graph D(k,q) proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is q-regular, edge-transitive and of girth at least k+4. For its exact girth g=g(D(k,q)), Füredi et al. (1995) conjectured g=k+5 for odd k and q≥4. This conjecture was shown to be valid in 2016 when (k+5)/2 is the product of an arbitrary factor of q−1 and an arbitrary power of the characteristic of Fq. In this paper, we determine all the girth cycles of D(k,q) for 3≤k≤5, q>3, and those for 3≤k≤14, q=3.