  An evolution algebra corresponds to a quadratic matrix $A$ of structural
constants. It is known the equivalence between nil, right nilpotent evolution
algebras and evolution algebras which are defined by upper triangular matrices
$A$. We establish a criterion for an $n$-dimensional nilpotent evolution
algebra to be with maximal nilpotent index $2^{n-1}+1$. We give the
classification of finite-dimensional complex evolution algebras with maximal
nilpotent index. Moreover, for any $s=1,...,n-1$ we construct a wide class of
$n$-dimensional evolution algebras with nilpotent index $2^{n-s}+1$. We show
that nilpotent evolution algebras are not dibaric and establish a criterion for
two-dimensional real evolution algebras to be dibaric.
