Abstract
The graph invariant $RM_2$, known under the name reduced second Zagreb index, is defined as $RM_2(G)=sum_{uvin E(G)}(d_G(u)-1)(d_G(v)-1)$, where $d_G(v)$ is the degree of the vertex $v$ of the graph $G$. In this paper, we give a tight upper bound of $RM_2$ for the class of graphs of order $n$ and size $m$ with at least one dominating vertex. Also, we obtain sharp upper bounds on $RM_2$ for all graphs of order $n$ with $k$ dominating vertices and for all graphs of order $n$ with $k$ pendant vertices. Finally, we give a sharp upper bound on $RM_2$ for all $k$-apex trees of order $n$. Moreover, the corresponding extremal graphs are characterized.
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