Why only the wavelength and speed of refracted light traveling inside a transparent material changes and not its frequency? When monochromatic light waves travel from one medium to another the frequency never changes.
A transition to a denser medium will result of a slow down of the propagation speed v of the light wave and its wavelength λ but not of its frequency f. Photons still travel at c speed from one atom to the next through the vacuum space between the atoms of the medium.
$$u=\lambda f$$
Should not both λ and f proportionally decrease to match the slower v speed?
Why f does not change what is the physical explanation?
Also, if f does not change and since no transparent material is perfect and there will be apart of reflection also some absorption, how light absorption, lost energy, is then justified by the equation
$$\mathrm{E}=\mathrm{h} \mathrm{f}$$
if f remains unchanged?
 A: There is a basic difference between "light" and the photon energy given by $E=hf$ where $f$ is the frequency of the classical electromagnetic wave emergent from a multitude of such photons.

Should not both λ and f proportionally decrease to match the slower f speed?

The way the photons, which always move with velocity c, build up the classical electromagnetic wave is not additive. It has been modeled  with quantum field theory and can be described only qualitatively in such an answer.
This single photon at a time double slit experiment can give an intuition of how individual photons with their individual paths build up a classical electromagnetic wave  (em) interference pattern.



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*Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.


The photons have individual paths with different lengths, which is what happens in a transparent medium too. Their speed will seem slower  because of the angular dispersion of the coherent photons inside the beam, they follow longer paths.

Why f does not change what is the physical explanation?

The individual  photon's energy given by the equation does not change , thus it will keep contributing to a classical em wave of frequency $f$

there will be apart of reflection also some absorption, how light absorption, lost energy, is then justified by the equation $E=hf$


if $f$ remains unchanged?

The equation is for individual photons, not for the classical em wave. Some individual photons will be lost from the beam, that is all.
A: The constant value of $f$ is easiest to understand by thinking about the wave model of light rather than the particle model.
If the frequency of the light wave inside and outside of a material had different values then there would be a discontinuity in the electric and magnetic fields at the boundary of the material. This is not physically realistic, so the frequency is constant, and speed and wavelength change in the same proportion as each other.
A: Due to continuity if one wavelength of light crosses a boundary, one must emerge on the other side. Hence the number of wavelengths crossing the boundary per second is equal to the number emerging per second. So the frequency of the incident and refracted wave must be the same.
For your second question, energy is lost by whole photons being absorbed/reflected, this is because they are a quanta of EM radiation, so cannot be subdivided.
