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

Abstract

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.