The first Zagreb index $M_{1}(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_{2}(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph $G$. This paper aims to investigate the Zagreb indices and coindices of the line graph and line cut-vertex graph of a class of unicyclic graphs called cycle-star graph using the notion of subdivision.