sorry, can't decode your png (ps embedded). And, from reading your post, I do not really understand your problem.

Unfortunately I cannot decipher your png file (embedded postscript). Could you provide a pdf version?

And from reading your post I do not understand your question. If you have a grid of k-points on which thephonon eigenvalue problem is solved then as a result you will get discrete solutions. If you have sufficient point density then you won't be able to discern this from a continuous curve... but I am probably missing your point here.

Dear Kai Fauth, I guess the reason why am not getting the dispersion right was due to the kpoint choice,. any professional advice to make the result better with the phonon dispersion attached.

Thanks for the picture. I have still not quite understood the nature of what you really consider to be your problem and how you expect this might be resolved with a different choice of k-points.

How to choose k-points is usually clear: you start with a real space lattice and get the reciprocal basis vectors with the usual definition. Then you construct the Wigner-Seitz cell in reciprocal space to obtain the first BZ. in the BZ you can identify points of particular symmetry, determine their coordinates and solve for the eigenvalues (phonons in your case) along the paths connecting these points. What is left for you to choose si the point density on these paths.

From your figure I guess that maybe you're intrigued by those places in the plot where dispersions approach each other and it might not be clear whether there is crossing or anti-crossing. Sometimes this can be seen when the point density is very high (in case of anti-crossing there is a gap at the crossing point), but in case the coupling is small it may require more points that you'd be willing to spend computing time for.

In case there is crossing (noninteracting dispersion branches) the eigenvalues of different branches do swap in the list of eigenvalues which you compute. Typical routines (Lapack,NAG) output an energy-sorted list of eigenvalues. In order to generate continous lines for crossing bands, you will have to swap their entries in the list of eigenvalues, so that you transit from a list of sorted eigenvalues at each k-point to a list of eigenvalues, sorted along a "band index". 

The computer code cannot decide by itself what to do, at leasit if only eigenvalues are computed. The character of the dispersion branches is encoded in the corresponding eigenvector. So, if what I think your problem might be is really what you're pondering, then you could/should analyze the eigenvectors and attribute band indices accordingly. Of yourse, interacting branches (avoided level crossing) change their character along the path between symmetry points.

Hope this is useful.

Dear Kai, I was able to identify the problem after several attempts, it was the high degeneracy in some high symmetry axis of the band structure that was a concern to me, which I felt was due to my choice of k point. I wish to inform you that it was resolved by setting the nbn  to old number (7,9,...)  to obtain the band structure. Any insight into the reason. see attached. I wish to fix the phonon dispersion too.

Dear Mustapha,

try xcrysden visualizer to generate k point in special path in BZ