On Zagreb eccentricity indices of cacti

•The effect of graft transformations to decrease and/or increase the Zagreb eccentricity indices is studied.•Sharp lower bounds on Zagreb eccentricity indices of graphs are established in C(n,k), where C(n,k) is the class of all cacti of order n with k cycles.•Sharp upper bounds on Zagreb eccentricity indices of graphs are established in C(n,k), where 0≤k≤⌊n−12⌋.•Corresponding extremal graphs are characterized. For a connected graph G, the first Zagreb eccentricity index is defined as ξ1(G)=∑u∈V(G)e2(u), and the second Zagreb eccentricity index is defined as ξ2(G)=∑uv∈E(G)e(u)e(v), where e(u) is the eccentricity of u in G. Let C(n,k) be the class of all cacti of order n with k cycles. In this paper, we establish sharp lower bounds on Zagreb eccentricity indices of graphs in C(n,k) and determine the corresponding extremal graphs. What’s more, we characterize the graphs in C(n,k) with maximal Zagreb eccentricity indices, where 0≤k≤⌊n−12⌋.