Coalescing Fiedler and core vertices

The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler verte...

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Bibliographic Details
Published in:Czechoslovak mathematical journal 2016-09, Vol.66 (3), p.971-985
Main Authors: Ali, Didar A., Gauci, John Baptist, Sciriha, Irene, Sharaf, Khidir R.
Format: Article
Language:English
Subjects:
Analysis
Apexes
Coalescing
Convex and Discrete Geometry
Eigenvalues
Graph theory
Insulators
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
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Summary:The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence G is determined relative to the nullity of G . This has direct applications in spectral graph theory as well as in the construction of certain ipso-connected nano-molecular insulators.
ISSN:0011-4642
1572-9141
DOI:10.1007/s10587-016-0304-8