Enumerative properties and cube polynomials of Tribonacci cubes

Tribonacci cubes ${\Gamma }_n^{\left(3\right)}$are induced subgraphs of $Q_n$, obtained by removing all the vertices that contain more than two consecutive 1’s. In the present work, we give some enumerative properties related to ${\Gamma }_n^{\left(3\right)}$. We show that the number of vertices of weight $w$ in ${\Gamma }_n^{\left(3\right)}$is ${\sum }_{j=0}^{n-w+1}\left(\genfrac{}{}{0ex}{}{n-w+1}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{w-j}\right)$ and express the number of edges of these graphs in terms of convolved Tribonacci numbers. We investigate the cube polynomials of Tribonacci cubes and determine the corresponding generating function. Finally, we give a formula for the number of induced $k$-cubes in ${\Gamma }_n^{\left(3\right)}$.