  The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF
exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is
studied by exact diagonalization (ED) and density matrix renormalization group
(DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary
conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap
$\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string
correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$
of consecutive spins in systems with two spins per unit cell; the ladder has
string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$. The minimum $N^*$ of
$g_2(N)$ is related to the range of ground-state spin correlations. Convergence
to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N
\geq N^*$. Singlet valence bond (VB) diagrams account for the size
dependencies. The frustrated ladder at special values of $J_F$, $J_L$ and $J_A$
reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and
the $J_1-J_2$ model, among others. Numerical analysis of ladders matches
previous results for spin-1 gaps or string correlation functions and extends
them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a
bond-order wave, a Kekul\'e VB diagrams at $J_L = J_F/2 \leq J_A$, that is
reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously
broken in the dimer phase of the $J_1-J_2$ model where the Kekul\'e diagrams
are the doubly degenerate ground states at $J_2/J_1 = 1/2$.
