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 \(Z{C_1}(G)={∑_{{v∈ V(G)}} {{τ _G}^2(v)} }\), \(ZC{}₂(G)=∑uv∈ E(G)^{}τ{}G(u)τ{}G(v)\) and \(ZC{}₁(G)=∑v∈ V(G)^{}d{}G(v)τ{}G(v)\), 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.