Minimal configuration trees |
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Authors: | Irene Sciriha Ivan Gutman |
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Institution: |
a University of Malta,
b University of Kragujevac, |
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Abstract: | 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. |
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Keywords: | Adjacency matrix Minimal configuration Core Periphery Interlacing Singular graphs Eigenvalues 2000 Mathematics Subject Classifications: 05C50 05C60 05B20 |
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