On the extremal cacti of given parameters with respect to the difference of zagreb indices

The first and the second Zagreb indices of a graph G are defined as $$M_1(G)= \sum _{v\in V_G}d_v^2 $$ M 1 ( G ) = ∑ v ∈ V G d v 2 and $$ M_2(G)= \sum _{uv\in E_G}d_ud_v$$ M 2 ( G ) = ∑ u v ∈ E G d u d v , where $$d_v,\, d_u$$ d v , d u are the degrees of vertices $$v,\, u$$ v , u in G. The difference of Zagreb indices of G is defined as $$\Delta M(G)=M_2(G)-M_1(G)$$ Δ M ( G ) = M 2 ( G ) - M 1 ( G ) . A cactus is a connected graph in which every block is either an edge or a cycle. Let $$\mathscr {C}_{n,k}$$ C n , k be the set of all n-vertex cacti with k pendant vertices and let $$\mathscr {C}_n^r$$ C n r be the set of all n-vertex cacti with r cycles. In this paper, the sharp upper bound on $$\Delta M(G)$$ Δ M ( G ) of graph G among $$\mathscr {C}_{n,k}$$ C n , k (resp. $$\mathscr {C}_n^r$$ C n r ) 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 $$\Delta M(G)$$ Δ M ( G ) of n-vertex cacti and n-vertex unicyclic graphs are determined, respectively. All the extremal graphs are characterized.