On maximum Zagreb connection indices for trees with fixed matching number

The first, second Zagreb connection indices and modified first Zagreb connection index are defined as \(Z{C_1}(G)={\sum\limits_{{v\in V(G)}} {{\tau _G}^2(v)} }\), \(ZC{}_{2}(G)=\displaystyle\sum_{uv\in E(G)}^{}\tau{}_{G}(u)\tau{}_{G}(v)\) and \(ZC{}_{1}^{\ast }(G)=\displaystyle\sum_{v\in V(G)}^{}d{}_{G}(v)\tau{}_{G}(v)\), respectively. In this paper, we consider the maximum values of \(Z{C_1}(G)\), \(Z{C_2}(G)\), \(Z{C_1}^{*}(G)\) of \(n\)-vertex trees with fixed matching number \(m\) and the extremal graphs are also characterized.