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You have two approaches to solve this exercise:

  1. Using Sagemath: generate all blocks (=biconnected graphs) of order k=3..n using graphs.nauty_geng("k -C"), then for each combination of 4 of them build a block graph with the required structure, compute its algebraic connectivity and record the maximum value.
  2. Using a sheet of paper: read carefully the definition of algebraic connectivity to understand what maximizes this value (hint: check the lower bounds), deduce the biconnected graph with maximum algebraic connectivity, use it to build the block graph you need, and finally prove that this graph answers your question (short proof). You can afterward use Sagemath to compute its algebraic connectivity.