Extremal Problems Involving the Two Largest Complementarity Eigenvalues of a Graph

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 $$\varrho (G)$$ ϱ ( G ) 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 $$\varrho _2(G)$$ ϱ 2 ( G ) 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 $$\varrho _2(G)$$ ϱ 2 ( G ) . 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 $$\varrho _2$$ ϱ 2 and $$\varrho -\varrho _2$$ ϱ - ϱ 2 on special classes of connected graphs. We establish several new results and propose a number of conjectures.