The Colin de Verdi\`ere parameter, excluded minors, and the spectral radius

Michael Tait

The Colin de Verdi\`ere parameter, excluded minors, and the spectral radius

Combinatorics 2018-12-18 v3

Authors: Michael Tait

Abstract

In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdi\`ere parameter at most $m$ . We also characterize graphs of maximum spectral radius with no $H$ as a minor when $H$ is either $K_r$ or $K_{s,t}$ . Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no $H$ as a minor when $r$ and $s$ are small, but not when they are larger.

Keywords

Colin de Verdière parameter excluded minor spectral radius adjacency matrix extremal graph

Cite

@article{arxiv.1703.09732,
  title  = {The Colin de Verdi\`ere parameter, excluded minors, and the spectral radius},
  author = {Michael Tait},
  journal= {arXiv preprint arXiv:1703.09732},
  year   = {2018}
}

Comments

To appear in Journal of Combinatorial Theory, Series A

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