2 deleted 97 characters in body
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k[] is function defined by you and tri is making UndirectedEdge-s

k[j_List] := Block[{t = j[[1]], r = j[[2]], i = j[[3]]},
  (2 t - 1) 2^(-i + r) 3^(-1 + i)]
tri[{t_, r_, i_}] := {
  {t, r, i} <-> {t, r + 1, i},
    {t, r + 1, i} <-> {t, r + 1, i + 1},
  {t, r + 1, i + 1} <-> {t, r, i}
 }

I remaked your code like this for making Graph.

d = 4;
data = Flatten@Table[tri[{1, i, j}],
    {i, 1, d - 1}, {j, 1, i}];
data2 = Apply[k[{##}] &, data, {2}];
vert = (VertexList@Graph@data)[[All, 2 ;; 3]];
co = Composition[
    RotationTransform[-\[Pi]/3],
    ShearingTransform[\[Pi]/6, {-1, 0}, {0, 1}], 
    ScalingTransform[Sqrt[3]/2, {0, 1}]] /@ vert;
Graph[data2,
 VertexLabels -> "Name",
 VertexCoordinates -> co
]co]

enter image description here

and I make number lines by following code, this is my guess you wanted.

l1 = k /@ VertexList@Graph@data;
l2 = Transpose@{l1, co};
l3 = Sort[l2, #1[[2, 1]] > #2[[2, 1]] && #1[[2, 2]] > #2[[2, 2]] &];
Reverse@(Transpose@l3)[[1]]

{1, 2, 3, 4, 6, 8, 9, 12, 18, 27}

PathGraph[Reverse@(Transpose@l3)[[1]],
  VertexLabels -> "Name",
 GraphLayout -> Automatic]"Name"]

Blockquote

9-row triangle

Blockquote

and this is number line of 9-row triangle

Blockquoteenter image description here

k[] is function defined by you and tri is making UndirectedEdge-s

k[j_List] := Block[{t = j[[1]], r = j[[2]], i = j[[3]]},
  (2 t - 1) 2^(-i + r) 3^(-1 + i)]
tri[{t_, r_, i_}] := {
  {t, r, i} <-> {t, r + 1, i},
  {t, r + 1, i} <-> {t, r + 1, i + 1},
  {t, r + 1, i + 1} <-> {t, r, i}
 }

I remaked your code like this for making Graph.

d = 4;
data = Flatten@Table[tri[{1, i, j}],
    {i, 1, d - 1}, {j, 1, i}];
data2 = Apply[k[{##}] &, data, {2}];
vert = (VertexList@Graph@data)[[All, 2 ;; 3]];
co = Composition[
    RotationTransform[-\[Pi]/3],
    ShearingTransform[\[Pi]/6, {-1, 0}, {0, 1}], 
    ScalingTransform[Sqrt[3]/2, {0, 1}]] /@ vert;
Graph[data2,
 VertexLabels -> "Name",
 VertexCoordinates -> co
]

enter image description here

and I make number lines by following code, this is my guess you wanted.

l1 = k /@ VertexList@Graph@data;
l2 = Transpose@{l1, co};
l3 = Sort[l2, #1[[2, 1]] > #2[[2, 1]] && #1[[2, 2]] > #2[[2, 2]] &];
Reverse@(Transpose@l3)[[1]]

{1, 2, 3, 4, 6, 8, 9, 12, 18, 27}

PathGraph[Reverse@(Transpose@l3)[[1]],
  VertexLabels -> "Name",
 GraphLayout -> Automatic]

Blockquote

9-row triangle

Blockquote

and this is number line of 9-row triangle

Blockquote

k[] is function defined by you and tri is making UndirectedEdge-s

k[j_List] := Block[{t = j[[1]], r = j[[2]], i = j[[3]]},(2 t - 1) 2^(-i + r) 3^(-1 + i)]
tri[{t_, r_, i_}] := {{t, r, i} <-> {t, r + 1, i},
    {t, r + 1, i} <-> {t, r + 1, i + 1},{t, r + 1, i + 1} <-> {t, r, i}}

I remaked your code like this for making Graph.

d = 4;
data = Flatten@Table[tri[{1, i, j}],{i, 1, d - 1}, {j, 1, i}];
data2 = Apply[k[{##}] &, data, {2}];
vert = (VertexList@Graph@data)[[All, 2 ;; 3]];
co = Composition[
    RotationTransform[-\[Pi]/3],
    ShearingTransform[\[Pi]/6, {-1, 0}, {0, 1}], 
    ScalingTransform[Sqrt[3]/2, {0, 1}]] /@ vert;
Graph[data2,VertexLabels -> "Name",VertexCoordinates -> co]

enter image description here

and I make number lines by following code, this is my guess you wanted.

l1 = k /@ VertexList@Graph@data;
l2 = Transpose@{l1, co};
l3 = Sort[l2, #1[[2, 1]] > #2[[2, 1]] && #1[[2, 2]] > #2[[2, 2]] &];
Reverse@(Transpose@l3)[[1]]

{1, 2, 3, 4, 6, 8, 9, 12, 18, 27}

PathGraph[Reverse@(Transpose@l3)[[1]], VertexLabels -> "Name"]

Blockquote

9-row triangle

enter image description here

1
source | link

k[] is function defined by you and tri is making UndirectedEdge-s

k[j_List] := Block[{t = j[[1]], r = j[[2]], i = j[[3]]},
  (2 t - 1) 2^(-i + r) 3^(-1 + i)]
tri[{t_, r_, i_}] := {
  {t, r, i} <-> {t, r + 1, i},
  {t, r + 1, i} <-> {t, r + 1, i + 1},
  {t, r + 1, i + 1} <-> {t, r, i}
 }

I remaked your code like this for making Graph.

d = 4;
data = Flatten@Table[tri[{1, i, j}],
    {i, 1, d - 1}, {j, 1, i}];
data2 = Apply[k[{##}] &, data, {2}];
vert = (VertexList@Graph@data)[[All, 2 ;; 3]];
co = Composition[
    RotationTransform[-\[Pi]/3],
    ShearingTransform[\[Pi]/6, {-1, 0}, {0, 1}], 
    ScalingTransform[Sqrt[3]/2, {0, 1}]] /@ vert;
Graph[data2,
 VertexLabels -> "Name",
 VertexCoordinates -> co
]

enter image description here

and I make number lines by following code, this is my guess you wanted.

l1 = k /@ VertexList@Graph@data;
l2 = Transpose@{l1, co};
l3 = Sort[l2, #1[[2, 1]] > #2[[2, 1]] && #1[[2, 2]] > #2[[2, 2]] &];
Reverse@(Transpose@l3)[[1]]

{1, 2, 3, 4, 6, 8, 9, 12, 18, 27}

PathGraph[Reverse@(Transpose@l3)[[1]],
 VertexLabels -> "Name",
 GraphLayout -> Automatic]

Blockquote

9-row triangle

Blockquote

and this is number line of 9-row triangle

Blockquote