The nature of glass transitions in chalcogenides and modified oxides depends on the network mean coordination number $\langle r\rangle$. These display systematic trends when spanning across the three topological phases: flexible, intermediate, and stressed-rigid. Trends in the glass-transition temperature Tg($\langle r\rangle$) show a monotonic increase with $\langle r\rangle$, but the nonreversing enthalpy of relaxation at Tg, ΔHnr($\langle r\rangle$), shows a deep- and square-well-like minimum with the walls representing the rigidity and stress transitions with increasing $\langle r\rangle$, respectively. In the well, the ΔHnr($\langle r\rangle$) term remains minuscule (∼0) corresponding to the isostatically rigid intermediate phase (IP). The melt fragility index (m) shows rather low values, m($\langle r\rangle$) < 20 for IP compositions, but increases outside the IP. Glass compositions in the IP show absence of network stress, form compacted networks, possess thermally reversing glass transitions, and display high glass-forming tendency—functionalities that have attracted widespread interest in understanding the physics of glasses and applications of the new IP formed.