LAUSR

The first and the second Zagreb indices of a graph G are defined as $$M_1(G)= \sum _{v\in V_G}d_v^2 $$M1(G)=∑v∈VGdv2 and $$ M_2(G)= \sum _{uv\in E_G}d_ud_v$$M2(G)=∑uv∈EGdudv, where $$d_v,\, d_u$$dv,du are the degrees of vertices $$v,\, u$$v,u in G. The difference of Zagreb indices of G is defined as $$\Delta M(G)=M_2(G)-M_1(G)$$ΔM(G)=M2(G)-M1(G). A cactus is a connected graph in which every block is either an edge or a cycle. Let $$\mathscr {C}_{n,k}$$Cn,k be the set of all n-vertex cacti with k pendant vertices and let $$\mathscr {C}_n^r$$Cnr be the set of all n-vertex cacti with r cycles. In this paper, the sharp upper bound on $$\Delta M(G)$$ΔM(G) of graph G among $$\mathscr {C}_{n,k}$$Cn,k (resp. $$\mathscr {C}_n^r$$Cnr) is established. Combining the results in Furtula et al. (Discrete Appl Math 178:83–88, 2014) and our results obtained in the current paper, sharp upper bounds on $$\Delta M(G)$$ΔM(G) of n-vertex cacti and n-vertex unicyclic graphs are determined, respectively. All the extremal graphs are characterized.