Unitary inner product graphs and their automorphisms. (English) Zbl 1521.05172
Summary: Let \(\mathbb{F}_{q^2}\) be a finite field of order \(q^2\) and \(2\nu +\delta\geq 2\) be an integer with \(\delta =0\) or \(1\), where \(q\) is a power of a prime. We introduce the concept of the unitary inner product graph \(Ui(2\nu +\delta, q^2)\) over \(\mathbb{F}_{q^2}\) and determine its automorphism group. We obtain two necessary and sufficient conditions for two vertices of \(Ui(2\nu +\delta, q^2)\) and two edges of \(Ui(2\nu +\delta, q^2)\), respectively, are in the same orbit under the action of the automorphism group of \(Ui(2\nu +\delta, q^2)\).
MSC:
05C76 | Graph operations (line graphs, products, etc.) |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
51F25 | Orthogonal and unitary groups in metric geometry |
15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
References:
[1] | Ashrafi, N., Maimani, H. R., Pournaki, M. R. and Yassemi, S., Unit graphs associated with rings, Comm. Algebra38(8) (2010) 2851-2871. · Zbl 1219.05150 |
[2] | Bondy, J. A. and Murty, U. S. R., Graph Theory, , Vol. 244 (Springer, New York, 2008). · Zbl 1134.05001 |
[3] | Cameron, P. J. and Ghosh, S., The power graph of a finite group, Discrete Math.311 (2011) 1220-1222. · Zbl 1276.05059 |
[4] | DeMeyer, F. R., McKenzie, T. and Schneider, K., The zero-divisor graph of a commutative semigroup, Semigroup Forum65(2) (2002) 206-214. · Zbl 1011.20056 |
[5] | Gu, Z. H. and Wan, Z. X., Orthogonal graphs of odd characteristic and their automorphisms, Finite Fields Appl.14 (2008) 291-313. · Zbl 1154.05061 |
[6] | Huang, L. P., Huang, Z. J., Li, C. K. and Sze, N. S., Graphs associated with matrices over finite fields and their endomorphisms, Linear Algebra Appl.447 (2014) 2-25. · Zbl 1288.05158 |
[7] | Li, F. G., Guo, J. and Wang, K. S., Orthogonal graphs over Galois rings of odd characteristic, Eur. J. Combin.39 (2014) 113-121. · Zbl 1284.05127 |
[8] | Li, F. G., Wang, K. S. and Guo, J., Symplectic graphs modulo \(pq\), Discrete Math.313 (2013) 650-655. · Zbl 1259.05194 |
[9] | Tang, Z. M. and Wan, Z. X., Symplectic graphs and their automorphisms, Eur. J. Combin.27 (2006) 38-50. · Zbl 1078.05092 |
[10] | Wan, Z. X., Geometry of Classical Groups over Finite Fields, 2nd edn. (Science Press, Beijing, New York, 2002). |
[11] | Wan, Z. X. and Zhou, K., Orthogonal graphs of characteristic 2 and their automorphisms, Sci. China Math.52 (2009) 361-380. · Zbl 1231.05294 |
[12] | Wan, Z. X. and Zhou, K., Unitary graphs and their automorphisms, Ann. Comb.14 (2010) 367-395. · Zbl 1235.05064 |
[13] | Wong, D., Ma, X. B. and Zhou, J. M., The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices, Linear Algebra Appl.460 (2014) 242-258. · Zbl 1300.05187 |
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.