The largest connected component in duplication-divergence growing graphs with symmetric coupled divergence is studied. Finite-size scaling reveals a phase transition occurring at a divergence rate ${\delta }_c$. The ${\delta }_c$ found is close to the locus of zero in Euler characteristic of finite-size graphs known to reflect the proximity of the largest connected component transition. A close correspondence with the vanishing of a scaling relation exponent for moments of the vertex degree distribution is shown, with such a scaling relation that generalizes a known form for duplication-divergence model graphs. The role of non-interacting vertices in shaping this transition with their presence or absence in duplication is also considered through a particular relation which would result in the two cases being comparable. The findings have relevancy for bond percolation in these growing graph models.