Hamiltonian Cycle on the Set of Genotypes and Non-Ergodic Quadratic Stochastic Operators

On the set of genotypes $\mathrm{\Phi }=\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ we introduce a binary relation generated by Volterra quadratic stochastic operator $V$ on $(m-1)$ dimensional simplex $S^{m-1}$ and prove that the operator $V$ be non-ergodic if either there exists a Hamiltonian cycle or one of the vertices $M_i=({\delta }_{1i},{\delta }_{2i},\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_{mi})$ of the simplex $S^{m-1}$ is a source and restriction of $V$ to the invariant face $F_i={x\in S^{m-1}: x_i=0}$isnon-ergodic\mathrm{.}Inthispaperweprovethisresultfor$m=2,3,4.$