Let $n \geq 1$ $n \geq 1$ and $U_{n}$ $U_{n}$ be the multiplicative group of the additive group $ℤ_{n}$ of integers modulo $n$ . The unitary Haar graph $H_{n} = Haar (ℤ_{n}, U_{n})$ is a graph with vertex set $ℤ_{n} \times {0, 1}$ and the edge set ${{(a, 0), (b, 1)} | b - a \in U_{n}}$ . In this paper, we study some structural properties of $H_{n}$ such as total chromatic number, diameter, planarity, the girth, vertex-connectivity and edge-connectivity. Also, some spectral properties of the graph $H_{n}$ are determined.

Communicated by Baoyindureng Wu

Keywords:

AMSC: 05C25

References

1. M. Arezoomand and B. Taeri , A classification of finite groups with integral bi-Cayley graphs, Trans. Comb. 4(4) (2015) 55–61. Google Scholar
2. M. Bašić and A. Ilić , Polynomials of unitary Cayley graphs, Filomat 29(9) (2015) 2079–2086. Crossref, Google Scholar
3. M. Boggess, T. Jackson-Henderson, I. Jiminez and R. Karpman , The structure of unitary Cayley graphs, SUMSRI J. (2008) 1–23. Google Scholar
4. A. Chandrashekar, E. Sampathkumar and M. A. Sriraj , Color energy of a unitary Cayley graph, Discuss. Math. Graph Theory 34 (2014) 707–721. Crossref, Google Scholar
5. I. Estélyi and T. Pisanski , Which Haar graphs are Cayley graphs?, Electron. J. Combin. 23(3) (2016) 3–10. Crossref, Google Scholar
6. Y.-Q. Feng, I. Kovács and D.-W. Yang , On groups all of whose Haar graphs are Cayley graphs, J. Algebraic Comb. 52 (2020) 59–76. Crossref, Google Scholar
7. C. Godsil and G. Royle , Algebraic Graph Theory, Graduate Texts in Mathematics, Vol. 207 (Springer, 2001). Crossref, Google Scholar
8. M. Hladnik, D. Marušič and T. Pisanski , Cyclic Haar graphs, Discrete Math. 244(1–3) (2002) 137–152. Crossref, Google Scholar
9. T. E. Huffman, Total coloring of graphs, Master’s thesis, San Jose State University (1989). Google Scholar
10. A. Ilić , The energy of unitary Cayley graphs, Linear Algebra Appl. 431(10) (2009) 1881–1889. Crossref, Google Scholar
11. W. Imrich and T. Pisanski , Multiple Kronecker covering graphs, Eur. J. Combin. 29(5) (2008) 1116–1122. Crossref, Google Scholar
12. W. Klotz and T. Sander , Some properties of unitary Cayley graphs, Electron. J. Combin. 14 (2007) 1–12. Crossref, Google Scholar
13. D. Waller , Double covers of graphs, Bull. Aust. Math. Soc. 14(2) (1976) 233–248. Crossref, Google Scholar
14. M. E. Watkins , Connectivity of transitive graphs, J. Comb. Theory 8(1) (1970) 23–29. Crossref, Google Scholar
15. D. B. West , Introduction to Graph Theory, 2nd edn. (Prentice-Hall, 2001). Google Scholar