For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E 3 (G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E 3 (G) = uv∈E(G) |ε G (u) − ε G (v)| and N(G) = {u,v}⊆V(G) |ε G (u) − ε G (v)|, where ε G (u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the non-self-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-self-centrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.