Laplacian Controllability for Graphs with Integral Laplacian Spectrum

Abstract

If G is a graph with n vertices, \(L_G\) is its Laplacian matrix, and \(\mathbf {b}\) is a binary vector of length n, then the pair \((L_G, \mathbf {b})\) is said to be controllable, and we also say that G is Laplacian controllable for \(\mathbf {b}\), if \(\mathbf {b}\) is non-orthogonal to any of the eigenvectors of \(L_G\). It is known that if G is Laplacian controllable, then it has no repeated Laplacian eigenvalues. If G has no repeated Laplacian eigenvalues and each of them is an integer, then G is decomposable into a (dominate) induced subgraph, say H, and another induced subgraph with at most three vertices. We express the Laplacian controllability of G in terms of that of H. In this way, we address the question on the Laplacian controllability of cographs and, in particular, threshold graphs.

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Acknowledgements

Research is partially supported by Serbian Ministry of Education, Science and Technological Development via University of Belgrade.

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Correspondence to Zoran Stanić.

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Stanić, Z. Laplacian Controllability for Graphs with Integral Laplacian Spectrum. Mediterr. J. Math. 18, 35 (2021). https://doi.org/10.1007/s00009-020-01684-3

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Keywords

  • Laplacian eigenvalues
  • Controllability
  • Integral Laplacian spectrum
  • Cograph
  • Threshold graph

Mathematics Subject Classification

  • 05C50
  • 93B05
  • 93C05