  An efficient density matrix renormalization group (DMRG) algorithm is
presented and applied to Y-junctions, systems with three arms of $n$ sites that
meet at a central site. The accuracy is comparable to DMRG of chains. As in
chains, new sites are always bonded to the most recently added sites and the
superblock Hamiltonian contains only new or once renormalized operators.
Junctions of up to $N = 3n + 1 \approx 500$ sites are studied with
antiferromagnetic (AF) Heisenberg exchange $J$ between nearest-neighbor spins
$S$ or electron transfer $t$ between nearest neighbors in half-filled Hubbard
models. Exchange or electron transfer is exclusively between sites in two
sublattices with $N_A \ne N_B$. The ground state (GS) and spin densities $
\rho_r = <S^z_r >$ at site $r$ are quite different for junctions with $S$ =
1/2, 1, 3/2 and 2. The GS has finite total spin $S_G = 2S (S)$ for even (odd)
$N$ and for $M_G =S_G$ in the $S_G$ spin manifold, $\rho_r > 0 (< 0)$ at sites
of the larger (smaller) sublattice. $S$ = 1/2 junctions have delocalized states
and decreasing spin densities with increasing $N$. $S$ = 1 junctions have four
localized $S_z = 1/2$ states at the end of each arm and centered on the
junction, consistent with localized states in $S$ = 1 chains with finite
Haldane gap. The GS of $S$ = 3/2 or 2 junctions of up to 500 spins is a spin
density wave (SDW) with increased amplitude at the ends of arms or near the
junction. Quantum fluctuations completely suppress AF order in $S$ = 1/2 or 1
junctions, as well as in half-filled Hubbard junctions, but reduce rather than
suppress AF order in $S$ = 3/2 or 2 junctions.
