This paper deals with various extremal problems involving two important parameters associated to a connected graph, say G. The first parameter is the largest complementarity eigenvalue: it is denoted by ϱ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\varrho (G)$$\end{document} and it is simply the spectral radius or index of the graph. Next in importance comes the second largest complementarity eigenvalue: it is denoted by ϱ2(G)\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _2(G)$$\end{document} and it is equal to the largest spectral radius among the children of G. By definition, a child or vertex-deleted connected subgraph of G is an induced subgraph obtained by removing a noncut vertex from G. In the first part of this work, we address the problem of identifying the eldest children and the youngest parents of G. We also analyze the uniqueness of such children and parents. An eldest child of G is a child whose spectral radius attains the value ϱ2(G)\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _2(G)$$\end{document}. The concept of youngest parent is somewhat dual to that of eldest child. The second part of this work is about minimization and maximization of the functions ϱ2\documentclass[12pt]{minimal}
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\begin{document}$$\varrho _2$$\end{document} and ϱ-ϱ2\documentclass[12pt]{minimal}
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\begin{document}$$\varrho -\varrho _2$$\end{document} on special classes of connected graphs. We establish several new results and propose a number of conjectures.
