Animating a graph changing edge colors

Specific Domains Graphs
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2

Your code does not generate the animation because it works with plt=Plots.plot(), but in your case plt=gplot() is of type Context.
To get it work you should install three (small) packages, Compose.jl, Cairo.jl and Fontconfig.jl, all three being necessary to transform each graph plot into a png file, with the function Compose.PNG:

using Graphs
using GraphPlot, Compose, Cairo, Fontconfig
using Random, Plots

let 
    G = Graph(5);
    edges =[(1,2), (2, 3), (1, 3), (1, 4), (4, 5), (1, 5)]
    for e in edges
       add_edge!(G, e)
    end        
 
    X = [1.0,0.0,0.0,2.0,2.0]
    Y = [0.5,0.0,1.0,0.0,1.0]
    cmap = range(HSL(colorant"blue"), stop=HSL(colorant"yellow"), length=360)
    colorlist = rand(100) * 360

    fname=String[]
    anim=Animation()
    for i=1:30
        color = cmap[Int(round(colorlist[i]))]
        edgecolor=["#389826","#CB3C33","#9558B4","#2458B2","#9558B2", color]
        plt= gplot(G, X,Y, nodelabel=1:5, NODESIZE=0.05, EDGELINEWIDTH=2,
                   nodefillc=colorant"grey", edgestrokec=edgecolor) 
        filename = joinpath(anim.dir, lpad(i, 6, "0")*".png")
        push!(fname, filename)
        draw(PNG(filename), plt)
    end
    anim = Animation(anim.dir, fname);
    Plots.buildanimation(anim, "anim-graph.gif", fps = 12, show_msg=false)
end