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Nullity of Graphs—A Survey and Some New Results

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Applied Linear Algebra, Probability and Statistics

Part of the book series: Indian Statistical Institute Series ((INSIS))

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Abstract

The spectrum of a graph G is the set of eigenvalues of the adjacency matrix of G. The nullity \(\eta (G)\), which is the algebraic multiplicity of the number zero in the spectrum of G, is a graph spectrum-based invariant. In the context of the H\(\overset{..}{\text {u}}\)ckel Molecular Orbital theory, the nullity of a molecular graph is used to determine the stability of unsaturated conjugated hydrocarbons. In this paper, we present a survey of significant results on nullity, such as bounds, transformations preserving nullity, nullity of line graphs, nullity versus energy and graphs with high nullity, along with some new results.

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Acknowledgements

Among the authors, Manjunatha Prasad Karantha wishes to acknowledge the funding support by Science and Engineering Research Board (DST), India, through the projects CRG/2019/000238 and MTR/2018/000156.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Ramco Institute of Technology, Rajapalayam, 626117, Tamil Nadu, India

    S. Arumugam

  2. Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India

    K. Arathi Bhat

  3. Faculty of Science, University of Kragujevac, Kragujevac, Serbia

    Ivan Gutman

  4. Department of Data Science, Prasanna School of Public Health, Manipal Academy of Higher Education, Manipal, India

    Manjunatha Prasad Karantha

  5. Centre for Advanced Research in Applied Mathematics and Statistics, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India

    Manjunatha Prasad Karantha

  6. Department of Data Science, Prasanna School of Public Health, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India

    Raksha Poojary

Authors
  1. S. Arumugam
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  2. K. Arathi Bhat
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  3. Ivan Gutman
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  4. Manjunatha Prasad Karantha
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  5. Raksha Poojary
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Corresponding author

Correspondence to S. Arumugam .

Editor information

Editors and Affiliations

  1. Indian Statistical Institute Delhi Center, New Delhi, India

    Ravindra B. Bapat

  2. Department of Data Science, CARAMS, Manipal Academy of Higher Education, Manipal, Karnataka, India

    Manjunatha Prasad Karantha

  3. Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada

    Stephen J. Kirkland

  4. Indian Statistical Institute Delhi Center, New Delhi, India

    Samir Kumar Neogy

  5. Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Assam, India

    Sukanta Pati

  6. Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland

    Simo Puntanen

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Arumugam, S., Arathi Bhat, K., Gutman, I., Karantha, M.P., Poojary, R. (2023). Nullity of Graphs—A Survey and Some New Results. In: Bapat, R.B., Karantha, M.P., Kirkland, S.J., Neogy, S.K., Pati, S., Puntanen, S. (eds) Applied Linear Algebra, Probability and Statistics. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-99-2310-6_8

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