  The hierarchical graphs and Sierpi\'nski graphs are constructed iteratively,
which have the same number of vertices and edges at any iteration, but exhibit
quite different structural properties: the hierarchical graphs are non-fractal
and small-world, while the Sierpi\'nski graphs are fractal and "large-world".
Both graphs have found broad applications. In this paper, we study consensus
problems in hierarchical graphs and Sierpi\'nski graphs, focusing on three
important quantities of consensus problems, that is, convergence speed, delay
robustness, and coherence for first-order (and second-order) dynamics, which
are, respectively, determined by algebraic connectivity, maximum eigenvalue,
and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of
Laplacian matrix. For both graphs, based on the explicit recursive relation of
eigenvalues at two successive iterations, we evaluate the second smallest
eigenvalue, as well as the largest eigenvalue, and obtain the closed-form
solutions to the sum of reciprocals (and square of reciprocals) of all nonzero
eigenvalues. We also compare our obtained results for consensus problems on
both graphs and show that they differ in all quantities concerned, which is due
to the marked difference of their topological structures.
