Kontsevich graphs associated with specific multiple zeta values

The primary aim of this paper is to provide an explicit construction of Kontsevich graphs whose integrals give certain multiple zeta values. Furthermore, by using this construction, we explicitly determine the weights of $\mathbb{Z}$-linear combinations of graphs whose integrals yield normalized multiple zeta values of a given weight. The methods employed in this paper are based on those introduced by Ritland, and our results provide a refinement of his results.