Geometric series of binary relation











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Let $rhosubset Xtimes X$ be a symmetric binary relation on a finite set $X$. Let $overline{rho}subset Xtimes X$ be its transitive closure :
$$
overline{rho}=bigcup_{i=0}^infty rho^{circ i}.
$$
If we choose a labeling $sigma:{1,dots,n}xrightarrowsim X$ of the elements of $X$ and consider the associated matrix $overline{M}=(overline{m}_{ij})$ whose entries are in ${0,1}$ and satisfy
$$
overline{m}_{ij}=begin{cases}
1 & text{if }(sigma(i),sigma(j))inoverline{rho},\
0 & text{if }(sigma(i),sigma(j))notinoverline{rho}
end{cases}
$$
then we get that if $L$ is the cardinality of the largest equivalence class in $overline{rho}$, and there are $c$ such equivalence classes, then
$$mathrm{Tr}(overline{M}^k)simeq cL^{k+1}$$




Question. Is there a procedure to extract $c$ and $L$ straight from $rho$ and its associated symmetric matrix ?







symmetric-matrices spectral-graph-theory







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    up vote
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    down vote

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    Let $rhosubset Xtimes X$ be a symmetric binary relation on a finite set $X$. Let $overline{rho}subset Xtimes X$ be its transitive closure :
    $$
    overline{rho}=bigcup_{i=0}^infty rho^{circ i}.
    $$
    If we choose a labeling $sigma:{1,dots,n}xrightarrowsim X$ of the elements of $X$ and consider the associated matrix $overline{M}=(overline{m}_{ij})$ whose entries are in ${0,1}$ and satisfy
    $$
    overline{m}_{ij}=begin{cases}
    1 & text{if }(sigma(i),sigma(j))inoverline{rho},\
    0 & text{if }(sigma(i),sigma(j))notinoverline{rho}
    end{cases}
    $$
    then we get that if $L$ is the cardinality of the largest equivalence class in $overline{rho}$, and there are $c$ such equivalence classes, then
    $$mathrm{Tr}(overline{M}^k)simeq cL^{k+1}$$




    Question. Is there a procedure to extract $c$ and $L$ straight from $rho$ and its associated symmetric matrix ?







    symmetric-matrices spectral-graph-theory







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $rhosubset Xtimes X$ be a symmetric binary relation on a finite set $X$. Let $overline{rho}subset Xtimes X$ be its transitive closure :
      $$
      overline{rho}=bigcup_{i=0}^infty rho^{circ i}.
      $$
      If we choose a labeling $sigma:{1,dots,n}xrightarrowsim X$ of the elements of $X$ and consider the associated matrix $overline{M}=(overline{m}_{ij})$ whose entries are in ${0,1}$ and satisfy
      $$
      overline{m}_{ij}=begin{cases}
      1 & text{if }(sigma(i),sigma(j))inoverline{rho},\
      0 & text{if }(sigma(i),sigma(j))notinoverline{rho}
      end{cases}
      $$
      then we get that if $L$ is the cardinality of the largest equivalence class in $overline{rho}$, and there are $c$ such equivalence classes, then
      $$mathrm{Tr}(overline{M}^k)simeq cL^{k+1}$$




      Question. Is there a procedure to extract $c$ and $L$ straight from $rho$ and its associated symmetric matrix ?







      symmetric-matrices spectral-graph-theory







      share|cite|improve this question













      Let $rhosubset Xtimes X$ be a symmetric binary relation on a finite set $X$. Let $overline{rho}subset Xtimes X$ be its transitive closure :
      $$
      overline{rho}=bigcup_{i=0}^infty rho^{circ i}.
      $$
      If we choose a labeling $sigma:{1,dots,n}xrightarrowsim X$ of the elements of $X$ and consider the associated matrix $overline{M}=(overline{m}_{ij})$ whose entries are in ${0,1}$ and satisfy
      $$
      overline{m}_{ij}=begin{cases}
      1 & text{if }(sigma(i),sigma(j))inoverline{rho},\
      0 & text{if }(sigma(i),sigma(j))notinoverline{rho}
      end{cases}
      $$
      then we get that if $L$ is the cardinality of the largest equivalence class in $overline{rho}$, and there are $c$ such equivalence classes, then
      $$mathrm{Tr}(overline{M}^k)simeq cL^{k+1}$$




      Question. Is there a procedure to extract $c$ and $L$ straight from $rho$ and its associated symmetric matrix ?







      symmetric-matrices spectral-graph-theory




      symmetric-matrices spectral-graph-theory






      share|cite|improve this question













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      Olivier Bégassat

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