In inferring stellar radius (R), what relationship governs L, R, and T?

L is proportional to R^2 T^4.

The relationship used to determine a star's radius ($R$) when its luminosity ($L$) and temperature ($T$) are known is derived from the Stefan-Boltzmann Law. This fundamental physical law states that the total energy radiated per unit area depends on the fourth power of the temperature, and the total luminosity depends on that area (which is related to the square of the radius). Therefore, Luminosity is proportional to the square of the Radius times the fourth power of the Temperature ($L \propto R^2 T^4$). Lines of constant radius on the H-R diagram follow this relationship, sloping from the upper-right toward the lower-left, allowing astronomers to visually compare the relative sizes of stars occupying different regions, such as massive Supergiants (large R) versus compact White Dwarfs (small R).