On the number of 7-cycles in regular n-tournaments

For a regular tournament TT of odd order nn, let cm(T)cm(T) be the number of cycles of length mm in TT. It is well known according to U. Colombo (1964) that c4(T)≤c4(RLTn), where RLTn is the unique regular locally transitive tournament of order nn. In turn, in 1968, A. Kotzig proved that c4(DRn)≤c4(T)c4(DRn)≤c4(T), where DRnDRn is a doubly-regular tournament of order nn. However, the spectral tools allow us to simply show that the converse inequality c5(RLTn)≤c5(T)≤c5(DRn) holds. Recently we have proved that c6(T)≤c6(DRn)c6(T)≤c6(DRn) and conjectured that c6(RLTn)≤c6(T). For these values of mm, the same results can be also formulated for the trace trm(T)trm(T) of the mmth power of the adjacency matrix of TT. (We consider this quantity here because it equals the number of closed walks of length mm in TT.) In the present paper, we determine c7(DRn)c7(DRn) and c7(RLTn). Comparing c7(DRn)c7(DRn) with c7(RLTn) yields the inequality c7(RLTn)