SOME GRAPH MAPPINGS THAT PRESERVE THE SIGN OF λ

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...

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Published inApplicable analysis and discrete mathematics Vol. 11; no. 1; pp. 148 - 165
Main Authors Mihailović, Bojana, Rašajski, Marija
Format Journal Article
LanguageEnglish
Published University of Belgrade 01.04.2017
Subjects
Cactus
Discrete mathematics
Eigenvalues
Equivalence relation
Graph theory
Linear algebra
Mathematical theorems
Polynomials
Spectral graph theory
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Summary: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 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.
ISSN:1452-8630
2406-100X