In Tikz, how to write equations on the top of foreach arrow?

Question

May you provide some guidance on how to add some descriptions on the arrow of the following two figures?
For example,
In the first figure,

can we add: Arrow $vec{21}$ on the top of arrow (better to be (1) parallel or (2) horizontal if possible, but not necessary if it is too difficult for you)  from the item 2 to the item 1?

Similarly, can we add: Arrow $vec{i j}$ on the top of each arrow from the item i to the item j? Here i=2, and j=1,3,4,5,6.

In the second figure,

can we try to add some greek alphabet, $alpha$, $beta$, $gamma$, etc on each of the arrow?

Can we make the above alignment to have the choices of being (1) parallel to the arrow or (2) horizontal to the document? If so, could you advise the command lines for each case?

Here is my Minimal Working note MWE:
documentclass{standalone}

usepackage{tikz}
usetikzlibrary{matrix}

usetikzlibrary{shapes,arrows,calc}
usepackage{multicol}
usepackage{multirow}
usepackage{amsmath, amsthm, amssymb,slashed,mathtools,tabu}

begin{document}

newline

;;;;
begin{tikzpicture}kern-5mm[>=stealth,->,shorten >=2pt,looseness=.5,auto]
      matrix (M)[matrix of math nodes,row sep=1cm,column sep=8mm]{
1 & 2 & 3
   4    &  5 & 6 
   };
       foreach a/b in {1-2/2-1,1-2/1-1,1-2/2-2,1-2/2-3, 1-2/1-3}
{draw[thick,->](M-a)--(M-b);}
    end{tikzpicture}

newline

begin{tikzpicture}[node distance=2cm, auto]

defmypoints{%
  (1, 0), (2, 0), (3, 0), (4, 0), (5, 0),
  (5, 1), (4, 1), (3, 1), (2, 1), (1, 1)%
};
path
  foreach x [count=xi] in mypoints {
    x node[circle, fill, inner sep=sqrt(2)*0.025cm] (nodexi) {}
  }
  foreach x [count=xi, remember=xi-1 as xiprev] in mypoints {
    ifnumxi>1 %
      (nodexiprev) edge[-latex', black!50!white] (nodexi)
    fi
  }
;
end{tikzpicture}

end{document}

The second figure is from and many thanks to https://tex.stackexchange.com/a/385800/41144

user213378 · Answer

TikZ can probably do it !
While waiting for an TikZ answer, I have an Asymptote answer as following:
Only for compare purpose!
usepackage("esvect");
picture pic1,pic2;

size(pic1,8cm);
pair[] A={(0,0),(-1,0),(1,0),sqrt(2)*dir(-135),(0,-1),sqrt(2)*dir(-45)};
int[] B={2,1,3,4,5,6};
margin m=Margin(2.5,2.5);
for (int i=0; i< A.length; ++i){  label(pic1,"$"+ (string) B[i] + "$",A[i],(0,0)); }
draw(pic1,Label("$vv{21}$"),A[0]--A[1],Arrow,m);
draw(pic1,Label("$vv{23}$",LeftSide),A[0]--A[2],Arrow,m);
draw(pic1,rotate(degrees(dir(A[0]-A[3])))*Label("$vv{24}$"),A[0]--A[3],Arrow,m);
draw(pic1,rotate(degrees(dir(A[0]-A[4])))*Label("$vv{25}$"),A[0]--A[4],Arrow,m);
draw(pic1,rotate(degrees(dir(A[5]-A[0])))*Label("$vv{26}$",LeftSide),A[0]--A[5],Arrow,m);

size(pic2,10cm);
pair[] A={(0,0),(1,0),(2,0),(3,0),(4,0),(4,1),(3,1),(2,1),(1,1),(0,1)};
string[] alphabet={"$alpha$", "$beta$", "$gamma$"};
alphabet.cyclic=true;
dot(pic2,A);
for (int i=0; i< A.length-1; ++i){
  draw(pic2,scale(0.8)*alphabet[i],A[i]--A[i+1],Arrow,Margin(1,1));
}
picture pic;
add(pic,pic1.fit(),(0,0),N);
add(pic,pic2.fit(),(0,0),S);
add(pic);
shipout(bbox(pic,2mm,invisible));

Zarko · Answer

With TikZ:
documentclass[tikz, margin=3mm]{standalone}
usetikzlibrary{arrows.meta,
                chains, 
                matrix,
                positioning,
                quotes}
usepackage{esvect}

begin{document}
    begin{tikzpicture}[
       every edge/.style = {draw, thick, -Straight Barb},
every edge quotes/.style = {auto, inner sep=2pt, sloped}
                        ] 
matrix (m) [matrix of math nodes,
             nodes={circle, minimum size=1.1em, inner sep=1pt, anchor=center},
             column sep=2cm,
             row sep=2cm]
{
1 & 2 & 3     
4 & 5 & 6     
};
foreach i/j/k in {1/1/1, 4/2/1, 5/2/2, 6/2/3, 3/1/3}
{
draw   (m-1-2) edge ["$vv{21}$"] (m-j-k);
}
    end{tikzpicture}

begin{tikzpicture}[
           node distance = 2cm and 2 cm,
             start chain = going right,
              dot/.style = {circle, fill, inner sep=2pt},
       every edge/.style = {draw=gray, thick, -Straight Barb},
every edge quotes/.style = {auto, inner sep=2pt, sloped}
                        ]

foreach i in {1,...,5}
{
    node (n1i) [dot, on chain] {};
    node (n2i) [dot, above=of n1i] {};
}
  foreach x [count=xi,count=xj from 2] in {alpha, beta, gamma, alpha}
ifnumxj<6     draw (n1xi) edge["$x$"] (n1xj);   fi
  foreach x [count=xi,count=xj from 2] in {gamma, beta, alpha,  gamma}
ifnumxj<6     draw (n2xj) edge["$x$"] (n2xi);   fi
draw (n15) edge["$beta$"] (n25);
 
    end{tikzpicture}
end{document}

In Tikz, how to write equations on the top of foreach arrow?

In the first figure,

In the second figure,

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