Concordance invariant $\mathrm{{\rm Y}}$ for balanced spatial graphs using grid homology

The $\mathrm{{\rm Y}}$ invariant is a concordance invariant defined by using knot Floer homology. Földvári gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\mathrm{{\rm Y}}$ invariant for balanced spatial graph using grid homology for balanced spatial graph. Regarding links as spatial graphs, we give a upper and lower bounds for $\mathrm{{\rm Y}}$ when two links are connected by a cobordism. Also we show that the combinatorial $\mathrm{{\rm Y}}$ is a concordance invariant for knots.