Minimal configuration trees

A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and e...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 54; no. 2; pp. 141 - 145
Main Authors Sciriha, Irene, Gutman, Ivan
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.03.2006
Subjects
2000 Mathematics Subject Classifications
Adjacency matrix
Core
Eigenvalues
Interlacing
Minimal configuration
Periphery
Singular graphs
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Summary:A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and edges, that has nullity one and the same core as G, is called a minimal configuration. A subdivision of a graph G is obtained by inserting a vertex on every edge of G. We review various properties of minimal configurations. In particular, we show that a minimal configuration is a tree if and only if it is a subdivision of some other tree.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081080500094055