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