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I am trying to implement the following algorithm into a Mathematica code but I am unsure how to proceed. Let $M$ be a 4x4 symmetric matrix, for example $$M=\begin{pmatrix} 1&1&0&1 \\ 1&0&0&1 \\ 0&0&1&0 \\ 1&1&0&0 \end{pmatrix} $$

  1. Go through the diagonal of the matrix $M$ and find the zero entries $e_{ij}$ first.
  2. In order of appearance, go through the row $i$ corresponding to $e_{ij}$ and find all 1 entries.
  3. Take the first '1' entry in column $k$ and check $e_{kk}$ in the diagonal. If $e_{kk}$=0, then stop and return $k$, else take the second column $l$ ($k$ < $l$) and check $e_{ll}$. If the corresponding diagonal entries in the $i$-th row are all 1, consider the second row and repeat the procedure.

In the example above, the algorithm first returns $M_{2,2}$ and $M_{4,4}$. First, it check the second row and finds $M_{2,1}$=1. It then checks that $M_{1,1}$=0 and thus it ignores it and goes to $M_{2,4}$=1. Then, it sees that $M_{4,4}$=0, thus it returns 4.

Edit: I already solved it and I will add the solution in case anyone is interested:

dzero[x_] := Flatten[Position[Diagonal[x], 0]];
k[x_] := Subsets[dzero[x], {2}]
p[x_] := Extract[k[x], FirstPosition[Extract[x, k[x]], 1]][[
  1]];
pp[x_] := 
 Extract[k[x], FirstPosition[Extract[x, k[x]], 1]][[
  2]];
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Maybe something like:

ClearAll[f]
f[m_] := Module[{i = 0, 
   p = DeleteDuplicates @ 
     Flatten[PositionIndex[m[[#]]]@ 1 & /@ PositionIndex[Diagonal @ m] @ 0]}, 
  Quiet @ Check[While[0 != m[[#, #]] &@p[[++i]]]; p[[i]], "failed"]]

Examples:

m = {{1, 1, 0, 1}, {1, 0, 0, 1}, {0, 0, 1, 0}, {1, 1, 0, 0}};
MatrixForm @ m

enter image description here

f[m]
 4
m2 = {{1, 1, 0, 1}, {1, 0, 0, 0}, {0, 0, 1, 0}, {1, 1, 0, 0}};
MatrixForm[m2]

enter image description here

f[m2]
2
m3 = {{1, 1, 0, 1}, {1, 0, 0, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}};
MatrixForm[m3]

enter image description here

f[m3]
"failed"

Update: An alternative approach:

ClearAll[f2]
f2[m_] := FirstCase[{a_, b_} /; m[[a, b]] == 1 :> b] @
  DeleteCases[{a_, a_}] @ Tuples[PositionIndex[Diagonal @ m] @ 0, 2]

f2 /@ {m, m2, m3}
 {4, 2, Missing["NotFound"]}
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