On disjoint union of $\mathrm{M}$-graphs

Given a pair $(X,\sigma)$ consisting of a finite tree $X$ and its vertex self-map $\sigma$ one can construct the corresponding Markov graph $\Gamma(X,\sigma)$ which is a digraph that encodes $\sigma$-covering relation between edges in $X$. $\mathrm{M}$-graphs are Markov graphs up to isomorphism. We obtain several sufficient conditions for the disjoint union of $\mathrm{M}$-graphs to be an $\mathrm{M}$-graph and prove that each weak component of $\mathrm{M}$-graph is an $\mathrm{M}$-graph itself.