An increase of the fundamental gap is indeed expected as you decrease the particle size. The keyword for thinking about this is "quantum confinement".

For free quantum objects, you will know that an increase in kinetic energy goes along with a reduction of the wavelength (details depending on the dispersion relation). [In 1D, the operator for the kinetic energy is proportional to the second spatial derivative.] Also from the standard quantum mechanical "particle in a box" problem you may be familiar with this situation.

Think of the nanoparticle as a potential well for both, electron and hole states. Asmaller particle forces the wave function of both electrons and holes to be more squeezed to "fit inside" (hence the curvature of the wave functions is larger and their kinetic energies larger, therefore).

To create an electron hole pair, you have to produce both states with increased energy and therefore the energy for creating the pair increases.

THis qualitative description relies on the assumption that but for the boundary contitions (= particle size) all other ingredients determining the properties of electron and hole states remain essentially unaltered. As you consider ever smaller particles, there will be a limit to this approximation.

 I should add that the above considerations are based on the "single particle" (electron, hole) energies. As the particle gets smaller, there is potentially stronger electron-hole-interaction in the excited state, leading to an increase of exciton binding energy or excitonic character of the absorption edge. This aspect would tend to a decrease of exciton excitation energy, and maybe also the relative oscillator strength (spectral weight) of absorption with excitonic character. This decrease is then superimposed on the increase of the single particle gap. Whether this is actually important in your material, I don't know. 

Dear Rajesh

Please read this article

Band gap narrowing in BaTiO3 nanoparticles facilitated by multiple mechanisms

S. Ramakanth and K. C. James Raju

J. Appl. Phys. 115, 173507 (2014); http://dx.doi.org/10.1063/1.4871776

Dear Rajesh

please read this article:

Effect of In concentration on .......and opt properties of In-doped ZnO nanostructures,in

Ceramics International 38(2012) 6295.

It is on Resarch gate.

Eg  increases with In addition

I belive that the origin of such high level transitions in a narrow band gap semiconductors like CdO can be due to the Burstein-Moss effect in semiconductor.  In the your case, the observed features  or 4 eV band gap may be attributed to the defects present in the CdO.

Thanks to all for giving your suggestions

Dear sir, The band gap is also depend on arrangement of atoms .