Unit Graphs Associated with Rings

Abstract

Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G(R) are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G(R) are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)

Key Words:

• Chromatic index
• Connectedness
• Diameter
• Girth
• Planarity
• Unit graph

2000 Mathematics Subject Classification:

• Primary 05C75
• Secondary 13A10, 13A15

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her helpful remarks which have contributed to improve the presentation of the article. The authors also express their gratitude to Professor Roger A. Wiegand for his kindness and support.

The research of N. Ashrafi was in part supported by a grant from Semnan University. The research of H. R. Maimani and M. R. Pournaki was in part supported by a grant from IPM (No. 87050213 and No. 87200111). The research of S. Yassemi was in part supported by a grant from the University of Tehran (No. 6103023/1/07).

Notes

Communicated by R. Wiegand.