The Laplacian matrix L for a graph G is the matrix L = D−A where D is the diagonal matrix of the vertex degrees and A is the traditional adjacency matrix. The Laplacian matrix is positive semidefinite with eigenvalues 0 = λ1 ≤ λ2 ≤ . . . ≤ λn. The eigenvalue λ2 is known as the algebraic connectivity a(G) of a graph. In this talk, we investigate the algebraic connectivity of maximal outerplanar graphs. We outline a proof that shows that if G is a maximal outerplanar graph on n ≥ 12 vertices, then a(G) ≤ 1 where equality holds on exactly two maximal outerplanar graphs on 12 vertices. The proof relies heavily on vertex labellings. (Received September 07, 2016)