This paper considered the concepts of monophonic, closed monophonic, and minimal closed monophonic numbers of a connected graph \(G\). It was shown that any positive integers \(m, n, d\), and \(k\) satisfying the conditions that \(4 \leq n \leq m, 3 \leq d \leq k\), and \(k \geq 2m – n + d + 1\) are realizable as the monophonic number, closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Also, any positive integers \(n, m, d\), and \(k\) with \(2 \leq n \leq m, d \geq 3\), and \(k \geq m + d – 1\) are realizable as the closed monophonic number, minimal closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Further, the closed monophonic number of the composition of connected graphs was also determined.