# A generalization of total graphs

## Abstract

Let *R* be a commutative ring with nonzero identity, \(L_{n}(R)\) be the set of all lower triangular \(n\times n\) matrices, and *U* be a triangular subset of \(R^{n}\), i.e., the product of any lower triangular matrix with the transpose of any element of *U* belongs to *U*. The graph \(GT^{n}_{U}(R^n)\) is a simple graph whose vertices consists of all elements of \(R^{n}\), and two distinct vertices \((x_{1},\dots ,x_{n})\) and \((y_{1},\dots ,y_{n})\) are adjacent if and only if \((x_{1}+y_{1}, \ldots ,x_{n}+y_{n})\in U\). The graph \(GT^{n}_{U}(R^n)\) is a generalization for total graphs. In this paper, we investigate the basic properties of \(GT^{n}_{U}(R^n)\). Moreover, we study the planarity of the graphs \(GT^{n}_{U}(U)\), \(GT^{n}_{U}(R^{n}{\setminus } U)\) and \(GT^{n}_{U}(R^n)\).

## Keywords

Total graph triangular subset planarity girth diameter## 2000 Mathematics Subject Classification

05C10 05C25 13A15## Notes

### Acknowledgements

The authors are grateful to the referee for careful reading of the manuscript and helpful suggestions.

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