Abstract
In this article we deal with the sign of ?2 - r, r > 0, where ?2 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 (?2 - r). These mappings induce equivalence relations among involved graphs, thus providing a way to indirectly apply the GRS-theorem (the generalization of so-called RS-theorem) to some GRS-undecidable (or RS-undecidable) graphs. To present possible applications, we revisit some of the previous results for reexive graphs (graphs whose second largest eigenvalue does not exceed 2). We show how maximal reexive graphs that belong to various families depending on their cyclic structure, can be reduced to RS-decidable graphs in terms of corresponding equivalence relations.
Highlights
The characteristic polynomial of a simple graph G on n vertices is defined by PG(λ) = det(λI − A), where A is (0, 1)-adjacency matrix of G
Some graph mappings that preserve the sign of λ2 − r
To illustrate some of the possible applications, we show how maximal reflexive graphs belonging to various families depending on their cyclic structure, determined in previous work [17, 18, 19], can be related to RS-decidable graphs with regard to the corresponding equivalence relations
Summary
By the Interlacing theorem, the property λ2 ≤ r is hereditary, i.e. if the second largest eigenvalue of a graph does not exceed r, the same holds for any of its subgraphs. Using these transformations we establish equivalence relations among the corresponding graphs. To illustrate some of the possible applications, we show how maximal reflexive graphs (i.e. those that do not admit reflexive extensions) belonging to various families depending on their cyclic structure, determined in previous work [17, 18, 19], can be related to RS-decidable graphs with regard to the corresponding equivalence relations
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