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Maximal and extremal singular graphs

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Abstract

A graph G is singular if the nullspace of its adjacency matrix is nontrivial. Such a graph contains induced subgraphs called singular configurations of nullity 1. We present two algorithms. One is for the construction of a maximal singular nontrivial graph G containing an induced subgraph, which is a singular configuration with the support of a vector in its nullspace as in that of G. The second is for the construction of a nut graph, a graph of nullity one whose null vector has no zero entries. An extremal singular graph of a given order, with the maximal nullity and support, has a nut graph as a maximal singular configuration.

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Authors and Affiliations

  1. Faculty of Science, University of Malta, Msida, Malta

    Irene Sciriha

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  1. Irene Sciriha
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Correspondence to Irene Sciriha.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 71, Algebraic Techniques in Graph Theory and Optimization, 2011.

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Sciriha, I. Maximal and extremal singular graphs. J Math Sci 182, 117–125 (2012). https://doi.org/10.1007/s10958-012-0733-3

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  • DOI: https://doi.org/10.1007/s10958-012-0733-3

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