The characteristic polynomials of symmetric graphs

Nafaa Chbili, Shamma Al Dhaheri, Mei Y. Tahnon, Amna A.E. Abunamous

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ, with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ[F] by one of Γ\ F. A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ.

Original languageEnglish
Article number582
JournalSymmetry
Volume10
Issue number11
DOIs
Publication statusPublished - Nov 2 2018

Keywords

  • Adjacency matrix
  • Automorphism group
  • Characteristic polynomial

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • Mathematics(all)
  • Physics and Astronomy (miscellaneous)

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