Some graph mappings that preserve the sign of λ2 - r

In this article we deal with the sign of λ2 − r, r > 0, where λ2 is the second largest eigenvalue of (adjacency matrix of) a simple graph and present some methods of determining it for some classes of graphs. The main result is a set of graph mappings that preserve the value of sgn (λ2 − r). These mappings induce equivalence relations among involved graphs, thus providing a way to indirectly apply the GRS-theorem (the generalization of so-called RStheorem) to some GRS-undecidable (or RS-undecidable) graphs. To present possible applications, we revisit some of the previous results for reflexive graphs (graphs whose second largest eigenvalue does not exceed 2). We show how maximal reflexive graphs that belong to various families depending on their cyclic structure, can be reduced to RS-decidable graphs in terms of corresponding equivalence relations.