Answer by kglr for How to find mean and median number of links of neighboring nodes

You can work directly with the input matrix

1. Construct a SparseArray from matrixlist
2. Use SparseArray`SparseArrayRemoveDiagonal to remove the diagonal
3. Take the property "AdjacencyLists"
4. Use MapIndexed+Association to get an association associating each vertex index to the indices of adjacent vertices

nL = Association @      MapIndexed[#2[[1]] -> # &] @       SparseArray`SparseArrayRemoveDiagonal[SparseArray @ matrixlist] @"AdjacencyLists"

<|1 -> {2, 7}, 2 -> {1, 5, 6, 9}, 3 -> {5, 6, 10}, 4 -> {},    5 -> {2, 3, 9, 10}, 6 -> {2, 3, 7, 10}, 7 -> {1, 6}, 8 -> {},    9 -> {2, 5}, 10 -> {3, 5, 6}|>

Map Length @* nL on nL (get the neighbor counts of neighbors):

nD = Map[Length @* nL] /@ nL<|1 -> {4, 2}, 2 -> {2, 4, 4, 2}, 3 -> {4, 4, 3}, 4 -> {},  5 -> {4, 3, 2, 3}, 6 -> {4, 3, 2, 3}, 7 -> {2, 4}, 8 -> {},  9 -> {4, 4}, 10 -> {3, 4, 4}|>

Map any function you desire on nD:

foo /@ nD

<|1 -> foo[{4, 2}], 2 -> foo[{2, 4, 4, 2}], 3 -> foo[{4, 4, 3}],  4 -> foo[{}], 5 -> foo[{4, 3, 2, 3}], 6 -> foo[{4, 3, 2, 3}],  7 -> foo[{2, 4}], 8 -> foo[{}], 9 -> foo[{4, 4}],  10 -> foo[{3, 4, 4}]|>

Mean /@ nD /. Mean[{}] -> {}

<|1 -> 3, 2 -> 3, 3 -> 11/3, 4 -> {}, 5 -> 3, 6 -> 3, 7 -> 3, 8 -> {},  9 -> 4, 10 -> 11/3|>

Median /@ nD /. Median[{}] -> {}

<|1 -> 3, 2 -> 3, 3 -> 4, 4 -> {}, 5 -> 3, 6 -> 3, 7 -> 3, 8 -> {},  9 -> 4, 10 -> 4|>

If you desire, you can replace key i with verticeLabel[[i]] in nD

nD2 = KeyMap[verticeLabel[[#]] &]@nD

<|"A1" -> {4, 2}, "A2" -> {2, 4, 4, 2}, "A3" -> {4, 4, 3}, "A4" -> {},"A5" -> {4, 3, 2, 3}, "B1" -> {4, 3, 2, 3}, "B2" -> {2, 4}, "B3" -> {}, "B4" -> {4, 4}, "B5" -> {3, 4, 4}|>