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This article is a continuation of our article in [Canad. J. Math. Vol. 72(3), (2020), pp. 774--804]. We construct orthogonal bases of the cycle and cutspaces of the Laakso graph $\mathcal{L}_n$. They are used to analyzeprojections from the edge space onto the cycle space and to obtain reasonablysharp estimates of the projection constant of$\operatorname{Lip}_0(\mathcal{L}_n)$, the space of Lipschitz functions on$\mathcal{L}_n$. We deduce that the Banach-Mazur distance fromTC$(\mathcal{L}_n)$, the transportation cost space of $\mathcal{L}_n$, to$\ell_1^N$ of the same dimension is at least $(3n-5)/8$, which is the analogueof a result from [op. cit.] for the diamond graph $D_n$. We calculate the exactprojection constants of $\operatorname{Lip}_0(D_{n,k})$, where $D_{n,k}$ is thediamond graph of branching $k$. We also provide simple examples of finitemetric spaces, transportation cost spaces on which contain $\ell_\infty^3$ and$\ell_\infty^4$ isometrically.

Analysis on Laakso graphs with application to the structure of transportation cost spaces