From statistics of regular tree-like graphs to distribution function and gyration radius of branched polymers

We consider flexible branched polymer, with quenched branch structure, and show that its conformational entropy as a function of its gyration radius $R$, at large $R$, obeys, in the scaling sense, $\Delta S \sim R^2/(a^2L)$, with $a$ bond length (or Kuhn segment) and $L$ defined as an average spanning distance. We show that this estimate is valid up to at most the logarithmic correction for any tree. We do so by explicitly computing the largest eigenvalues of Kramers matrices for both regular and "sparse" 3-branched trees, uncovering on the way their peculiar mathematical properties.