Given a connected graph G, we say that a set EC;{subseteq};V(G)$ is convex in G if, for every pair of vertices x, $y;{in};C$, the vertex set of every x - y geodesic in G is contained in C. The convexity number of G is the cardinality of a maximal proper convex set in G. In this paper, we show that every pair k, n of integers with $2;{leq}k;{leq};n;-;1$ is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number of k-regular graphs of order n with n > k+1.