Creating hexagonal grid (hexagonal grid graph)

With IGraph/M:

IGMeshGraph@IGLatticeMesh["Hexagonal", {6, 4}]

We can also crop it to a hexagon:

IGMeshGraph@IGLatticeMesh["Hexagonal", Polygon@CirclePoints[10, 6]]

It can also generate many other kinds of lattices, not just hexagonal.

Here is my generalization of the code from link provided by @LouisB:

HexagonalGridGraph2[{wide1_Integer?Positive, wide2_Integer?Positive, 
   wide3_Integer?Positive}, opts : OptionsPattern[Graph]] := 
 Module[{cells, edges, vertices}, 
  cells = 
   Flatten[Table[
     CirclePoints[{Sqrt[3] (1 j + k - 2 ) + Sqrt[3] (1 j + l - 2 ), 
       3 k - 2 - 3 l}, {2, \[Pi]/2}, 6], {j, wide1}, {k, wide2}, {l, 
      wide3}], 2];
  edges = Union[Sort /@ Flatten[Partition[#, 2, 1, 1] & /@ cells, 1]];
  vertices = Union[Flatten[edges, 1]];
  IndexGraph[
   Graph[UndirectedEdge @@@ edges, opts, 
    VertexCoordinates -> Thread[vertices -> vertices]]]]

And here are some examples:

Sort /@ Tuples[Range[4], {3}] // Union;
Partition[
  Rasterize /@ (HexagonalGridGraph2[#, PlotLabel -> #, 
       ImageSize -> {100, 100}] & /@ %), 5];
ImageAssemble[%]

We can generate the vertex coordinates using a slightly modified version of azerbajdan's cells and use them with NearestNeighborGraph:

ClearAll[vCoords]
vCoords = DeleteDuplicates @ Flatten[
   Table[CirclePoints[{(2 j + k + l - 4) Sqrt[3] , 3 k - 2 - 3 l}, {2, π/2}, 6], 
     {j, #}, {k, #2}, {l, #3}], 3] &;

ClearAll[hexGridGraph]
hexGridGraph = Module[{v = vCoords @@ #},
   NearestNeighborGraph[v, ##2, VertexCoordinates -> v]] &;

Examples:

hexGridGraph[{3, 5, 7}, 
  VertexLabels -> Placed["Index", Center], 
  VertexSize -> .7, 
  VertexStyle -> White, 
  VertexLabelStyle -> 8, 
  ImageSize -> 400]

args = Sort /@ Tuples[Range[4], {3}] // Union;

hexGridGraph[#, PlotLabel -> #, ImageSize -> {100, 100}] & /@ args // 
    Multicolumn[#, 5] &