  In this paper, we prove the first-order convergence law for the uniform
attachment random graph with almost all vertices having the same degree. In the
considered model, vertices and edges are introduced recursively: at time $m+1$
we start with a complete graph on $m+1$ vertices. At step $n+1$ the vertex
$n+1$ is introduced together with $m$ edges joining the new vertex with $m$
vertices chosen uniformly from those vertices of $1,\ldots,n$, whom degree is
less then $d=2m$. To prove the law, we describe the dynamics of the logical
equivalence class of the random graph using Markov chains. The convergence law
follows from the existence of a limit distribution of the considered Markov
chain.
