Let $\rho\subset X\times X$ be a symmetric binary relation on a finite set $X$. Let $\overline{\rho}\subset X\times X$ be its transitive closure : $$ \overline{\rho}=\bigcup_{i=0}^\infty \rho^{\circ i}. $$ If we choose a labeling $\sigma:\{1,\dots,n\}\xrightarrow\sim X$ of the elements of $X$ and consider the associated matrix $\overline{M}=(\overline{m}_{ij})$ whose entries are in $\{0,1\}$ and satisfy $$ \overline{m}_{ij}=\begin{cases} 1 & \text{if }(\sigma(i),\sigma(j))\in\overline{\rho},\\ 0 & \text{if }(\sigma(i),\sigma(j))\notin\overline{\rho} \end{cases} $$ 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 ?