adjMat1 = {{Infinity, 1, 1, 1, 1},
{1, Infinity, 1, 1, 1},
{1, 1, Infinity, 1, 1},
{1, 1, 1, Infinity, 1},
{1, 1, 1, 1, Infinity}};

adjMat2 = {{Infinity, 8, 6, 3, 7},
{1, Infinity, 1, 1, 1},
{2, 5, Infinity, 9, 4},
{1, 1, 1, Infinity, 1},
{1, 1, 1, 1, Infinity}};

WeightedAdjacencyGraph[adjMat1, DirectedEdges -> True]
WeightedAdjacencyGraph[adjMat2, DirectedEdges -> True]

adjMat1 = {{Infinity, 1, 1, 1, 1}, {1, Infinity, 1, 1, 1}, {1, 1,
    Infinity, 1, 1}, {1, 1, 1, Infinity, 1}, {1, 1, 1, 1, Infinity}};

adjMat2 = {{Infinity, 8, 6, 3, 7}, {1, Infinity, 1, 1, 1}, {2, 5,
    Infinity, 9, 4}, {1, 1, 1, Infinity, 1}, {1, 1, 1, 1, Infinity}};

multiedge = DeleteCases[ Flatten@Table[
    If[i == j, Null, Table[i -> j, {adjMat2[[i, j]]}]], {i,
     Length[adjMat2]}, {j, Length@First@adjMat2}], Null]

(* Output:   {1 -> 2, 1 -> 2, 1 -> 2, 1 -> 2, 1 -> 2, 1 -> 2, 1 -> 2, 1 -> 2,
 1 -> 3, 1 -> 3, 1 -> 3, 1 -> 3, 1 -> 3, 1 -> 3, 1 -> 4, 1 -> 4,
 1 -> 4, 1 -> 5, 1 -> 5, 1 -> 5, 1 -> 5, 1 -> 5, 1 -> 5, 1 -> 5,
 2 -> 1, 2 -> 3, 2 -> 4, 2 -> 5, 3 -> 1, 3 -> 1, 3 -> 2, 3 -> 2,
 3 -> 2, 3 -> 2, 3 -> 2, 3 -> 4, 3 -> 4, 3 -> 4, 3 -> 4, 3 -> 4,
 3 -> 4, 3 -> 4, 3 -> 4, 3 -> 4, 3 -> 5, 3 -> 5, 3 -> 5, 3 -> 5,
 4 -> 1, 4 -> 2, 4 -> 3, 4 -> 5, 5 -> 1, 5 -> 2, 5 -> 3, 5 -> 4} *)

Manipulate[
 Row[{GraphPlot[multiedge, MultiedgeStyle -> None, Method -> s],
   GraphPlot[adjMat1, MultiedgeStyle -> None, Method -> s],
   AdjacencyGraph[adjMat1 /. [Infinity] -> 0,
    GraphLayout -> s]}], {s, {"SpringElectricalEmbedding",
   "SpringEmbedding", "HighDimensionalEmbedding", "CircularEmbedding",
    "RandomEmbedding", "LinearEmbedding"}}]

Options[VisualizeNetwork] = Options[Graph];
VisualizeNetwork[{adjMat2_, edgecoloring_: {Green, Red},
scaling_: 600, opacity_: 0.5, ShowEdgeWeight_: True},
opts : OptionsPattern[]] :=
Block[{gr, weight, Nweight, edge, nodes, vertexStyle, EdgeBlendColor,edgeStyle, g},
gr = WeightedAdjacencyGraph[adjMat2, DirectedEdges -> True];
(*Lets extract the edge weight list from your graph object*)
weight = AbsoluteOptions[gr, EdgeWeight][[1, 2]];
(*Extract the edge list*)
edge = EdgeList@gr;
(*Extract the vertex list*)
nodes = VertexList@gr;
(* Make a color blend by scaling all the edge weight with the maximum edge weight *)
EdgeBlendColor = Map[ Blend[edgecoloring, #/Max[weight]] &, weight];
(* Make a thickness by scaling all the edge weight with the maximum edge weight *)
edgeStyle =
MapThread[(#1 -> {Opacity[opacity], #3,
     Thickness[#2/scaling]}) &, {edge, weight, EdgeBlendColor}];
Nweight = MapThread[
 Rasterize[#1, ImageSize -> 9,
   Background -> Lighter@#2] &, {weight, EdgeBlendColor}];
 g = If[ShowEdgeWeight == True,
 Graph[nodes, edge, EdgeStyle -> edgeStyle,
  EdgeLabels -> Flatten@MapThread[#1 -> #2 &, {edge, Nweight}],
  VertexLabels ->
   Table[i -> Placed["Name", {1/2, 1/2}], {i, nodes}], opts],
 Graph[nodes, edge, EdgeStyle -> edgeStyle,
  VertexLabels ->
   Table[i -> Placed["Name", {1/2, 1/2}], {i, nodes}], opts]
 ];
g
];

adjMat1 = {{Infinity, 1, 1, 1, 1}, {1, Infinity, 1, 1, 1}, {1, 1,
Infinity, 1, 1}, {1, 1, 1, Infinity, 1}, {1, 1, 1, 1, Infinity}};
adjMat2 = {{Infinity, 8, 6, 3, 7}, {1, Infinity, 1, 1, 1},
{2, 5,Infinity, 9, 4}, {1, 1, 1, Infinity, 1}, {1, 1, 1, 1, Infinity}};
(* Some Graph visualization option *)
opts = {VertexSize -> Medium, VertexSize -> Medium,
VertexStyle ->Directive[Opacity[0.65`], Blue, EdgeForm[None]],
VertexLabelStyle -> Directive[GrayLevel[0], 12],
VertexSize -> {"Scaled", 0.3`},
GraphLayout -> "SpringElectricalEmbedding"};
(* Call the function *)
GraphicsGrid@{{VisualizeNetwork[{adjMat1, {Yellow, Red}, 600, 0.5,
 True}, opts],
VisualizeNetwork[{adjMat2, {Yellow, Red}, 600, 0.5, True}, opts]}}

gr = WeightedAdjacencyGraph[adjMat2, DirectedEdges -> True];
weight = AbsoluteOptions[gr, EdgeWeight][[1, 2]];
edge = EdgeList@gr;
k = Graph[edge, EdgeLabels -> (el=MapThread[#1 -> #2 &, {edge, weight}])];
g = (AbsoluteOptions[k,
  EdgeLabels] /. {(a_ [DirectedEdge] b_ -> c_) :>  {a -> b,
    c}})[[1, 2]];

GraphPlot[g,
   EdgeRenderingFunction -> ({Text[Style[#3, 15], Mean[#1]], Blue,
   AbsoluteThickness[0.5 + #3/5], Arrowheads[0.02 + #3/170],
   Arrow[#1, 0.075]} &), VertexLabeling -> True]

{e, w} = Transpose[g]
Graph[e, EdgeWeight -> w,
EdgeLabelStyle -> Directive[Red, 20, Background -> White],
EdgeLabels -> el ,
EdgeShapeFunction ->
   el /. {((a_ [DirectedEdge] b_) ->  c_) -> ((a [DirectedEdge]
    b) -> ({AbsoluteThickness[.9 + c/4], Arrow[#1, 0.025]} &))}]

Graph[{1 <-> 2, 2 <-> 3}, EdgeWeight -> {1, 2},
  GraphLayout -> {"SpringElectricalEmbedding", "EdgeWeighted" -> True}]

<<IGraphM`
g = Graph[{1 <-> 2, 2 <-> 3}, EdgeWeight -> {1, 2}];
IGLayoutFruchtermanReingold[g]