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
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
add a comment |
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
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
add a comment |
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
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
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
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
asked Nov 19 at 14:59
Olivier Bégassat
13.2k12273
13.2k12273
add a comment |
add a comment |
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