multiplicative zagreb eccentricity indices of some composite graphs

let $g$ be a connected graph. the multiplicative zagreb eccentricity indices of $g$ are defined respectively as ${bf pi}_1^*(g)=prod_{vin v(g)}varepsilon_g^2(v)$ and ${bf pi}_2^*(g)=prod_{uvin e(g)}varepsilon_g(u)varepsilon_g(v)$, where $varepsilon_g(v)$ is the eccentricity of vertex $v$ in graph $g$ and $varepsilon_g^2(v)=(varepsilon_g(v))^2$. in this paper, we present some bounds of the multiplicative zagreb eccentricity indices of cartesian product graphs by means of some invariants of the factors and supply some exact expressions of ${bf pi}_1^*$ and ${bf pi}_2^*$ of some composite graphs, such as the join, disjunction, symmetric difference and composition of graphs, respectively.