×

Singular graphs with dihedral group action. (English) Zbl 1455.05043

Summary: Let \(\Gamma\) be a simple undirected graph on a finite vertex set and let \(A\) be its adjacency matrix. Then \(\Gamma\) is singular if \(A\) is singular. The problem of characterizing singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Software:

OEIS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] AL-Tarimshawy, A., Singular graphs (2018), University of East Anglia: University of East Anglia Norwich UK, (Ph.D thesis)
[2] Babai, L., Spectra of Cayley graphs, J. Combin. Theory Ser. B, 27, 2, 180-189 (1979) · Zbl 0338.05110
[3] Bass, H.; Estes, D. R.; Guralnick, R. M., Eigenvalues of symmetrical matrices and graphs, J. Algebra Volume, 168, 2, 536-567 (1979) · Zbl 0826.15006
[4] von Collatz, L.; Sinogowitz, U., (Spektren endlicher Grafen. Spektren endlicher Grafen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 21, 1 (1957), Springer-Verlag), 63-77 · Zbl 0077.36704
[5] Davis, P. J., Circulant Matrices (1979), John Wiley & Sons: John Wiley & Sons New York · Zbl 0418.15017
[6] Godsil, C.; Royle, G. F., Algebraic Graph Theory, Vol. 207 (2013), Springer Science & Business Media
[7] Graovac, A.; Gutman, I.; Trinajstić, N.; Ivković, T., Graph theory and molecular orbitals, Theor. Chim. Acta, 26, 1, 67-78 (1972)
[8] Kra, I.; Simanca, S. R., On circulant matrices, Notices Amer. Math. Soc., 59, 3, 368-377 (2012) · Zbl 1246.15030
[9] Lal, A. K.; Reddy, A. S., Non-singular circulant graphs and digraphs (2011), arXiv preprint arXiv:1106.0809
[10] Liu, X.; Zhou, S., Eigenvalues of cayley graphs (2018), arXiv preprint arXiv:1809.09829
[11] Müller, J.; Neunhöffer, M., Some computations regarding Foulkes’ conjecture, Experiment. Math., 14, 3, 277-283 (2005) · Zbl 1081.05106
[12] The Online Encyclopedia of Integer Sequences, https://oeis.org. · Zbl 1274.11001
[13] Siemons, J.; Zalesski, A., Remarks on singular Cayley graphs and vanishing elements of simple groups, J. Algebraic Combin., 50, 4, 379-401 (2019) · Zbl 1429.05093
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.