Thanks a lot Henrik and Horvát:
Both are working. However, the number of cells/connectivities, in case of using MeshCells[mesh, 2], is giving 42 connectivity:
{{12, 11, 2}, {11, 12, 4}, {12, 2, 4}, {2, 11, 4}, {9, 3, 10}, {3, 9, 7}, {9, 10, 7}, {10, 3, 7}, {5, 6, 12}, {6, 5, 10}, {5, 12, 10},
{12, 6, 10}, {12, 9, 5}, {9, 12, 10}, {5, 9, 10}, {12, 8, 2}, {8, 12, 7}, {12, 2, 7}, {2, 8, 7}, {9, 5, 11}, {5, 12, 11}, {12, 9, 11},
{3, 10, 1}, {10, 9, 1}, {9, 3, 1}, {10, 5, 1}, {5, 9, 1}, {10, 6, 1}, {6, 5, 1}, {11, 2, 7}, {12, 11, 7}, {3, 8, 10}, {8,3, 7}, {10, 8, 7},
{11, 5, 4}, {5, 12, 4}, {5, 6, 4}, {6, 12,4}, {12, 9, 7}, {9, 11, 7}, {12, 10, 7}, {10, 8, 12}}
and the number of triangles on the surface for 12-vertex is 20.
I am using the command (MeshCells) to display the vertex indices to understand what are the extra triangles.
I anticipated these results are due to making all possible connectivities of tree vertices without restricting these triangular elements on the surface of the sphere.
Thank you very much