There is no general formula in terms of radicals for a general 5th degree polynomial. For some cases of coefficients there will be solutions, roots(x^5-1); is a simple example.

As @Carl Love says, walks are enumerated by the adjacency matrix, and can be used to do this.

Download Walks3.mw

I think this is fairly robust.

get_exponent:=(expr,x::name)->diff(expr,x)/expr*x:

get_exponent(x^2,x);
get_exponent(x^(n-1),x);
get_exponent(3*x^(n+2),x);
get_exponent(3*x[2]^k*x[3]^(n+2),x[3]);

    
                               2

                             n - 1

                             n + 2

                             n + 2

Once you entered floating point values in your Matrix, the implication is that you have given up on symbolic solutions. Since the Matrix wasn't obviously of a structure that has real eigenvalues, the algorithm used was a general one that generated complex values. As @acer pointed out, you can then make them look nicer, but the algorithm used is the same. For a symmetric Matrix, if you use shape=symmetric, then Maple will choose an algorithm that will produce real eigenvalues and eigenvectors (try it on A:=Matrix(2,2,[[0.8,0.3],[0.3,0.7]],shape=symmetric);).

But if you enter exact values in the first place, then Maple will do the whole calculation exactly. In general, this is the power of Maple over a regular programming language. Here I converted the values you entered to rationals to avoid re-entering, but you could have entered these directly.

Download Eigenvectors.mw

Yes, my Maple 2023.0 on Windows 10 shows this behaviour.

Try Insert->Table and choose 1 row, 1 column, Title: "Fig. 1" and tick Show caption. Then Insert->Image->From file and choose the file. After you have it, right-click (cmd-click on Mac) and choose Table -> Properties, Table size 50% and the image will reduce size 50% keeping the aspect ratio. That menu also has a "Caption" tab which gives several ways to specify the caption. As far as I know there isn't automatic renumbering.

(As you probably already found, you can insert images in a text area but resizing with the handles doesn't maintain the aspect ratio.)

(Notice that _B2 stands for a binary variable with values 0 or 1, but _Z1 and _Z2 are (different) integers.)

Download underline.mw

Your code has a list M := [0, 0.5, 1.0, 1.5, 5] which you insert into other expressions as though it were a single number. So even for F[3] that isn't sensible. This ultimately leads to the error message.

mul(map(mul,ListTools:-Collect(A)))

CN and N1 are 

CN := (G, u) -> add(map2(GraphTheory:-Degree, G, GraphTheory:-Neighborhood(G, u, 'closed')));
N1 := G -> add(map(e->(CN(G,e[1])+CN(G,e[2]))/2, convert(GraphTheory:-Edges(G),list)));

The others are straightforward modifications.

This gives the answer in term of four algebraic numbers, each of which can be written in terms of the first root, which I think is more in the spirit of Galois theory than radicals. I also show the minimal polynomials that show the Galois group is order 72, But I'm not using the Galois group for much of this, as the OP asked for. Sorry if this is already obvious to the OP; perhaps we can have some clarification if the answers here are off track.

Download Splits.mw

That is a surprising result, and sufficiently far enough from the correct answer that I would call it a bug, even though it is just a numerical issue. Certainly since the matrix is "nice" (symmetric, diagonally dominant). As @vv says it depends on the precision, but certainly you don't need to go to 40 to get OK results. At Digits=14, M_MP.M_MP - C is recognizably close to zero and at Digits=16, the entries are all about 1e-6 or 1e-7.

In the general case, the algorithm for matrix functions is nontrivial, but Maple often has special cases for shape=symmetric or shape = hermitian, so then I think just applying the function to the eigenvalues as you did is straightforward.

@vv's comment about the determinant might be true (I agree), but it seems to imply that the reason that <0,1;0,0> doesn't have a square root is because its determinant is zero. That has more to do with the fact that it is non-diagonalizable; for example <1,0;0,0> (positive semidefinite) and <1,1;0,0> both have square roots and have determinant zero. 

For the edge separator, Maple's IsCutSet can be used - it just checks for an increase in the number of components. Checking for a Cut is more complicated since just checking for 2 components can fail to detect extra removed edges that are in one of the components. Here is one way to do it.

[Edit - this actually tests for a cutset, which is a minimal Cut, so I've renamed it IsMinCut; see below for the test for a Cut]

Download IsMinCut.mw

It looks as though when there are more unkowns than equations that Mathematica sets the extra ones first in order to be zero. So something like this achieves that result.

Download solve.mw

There are many ways; here is one. (Note that CycleBasis doesn't give all cycles in the graph)

Download countcycles.mw