A GENERALIZATION OF TOTAL GRAPHS

Proceedings of the Indian Academy of Sciences. Mathematical sciences

M AFKHAMI^1¹  K HAMIDIZADEH^2² 

[1] Department of Mathematics, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran^1;Department of Pure Mathematics, International Campus of Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran^2

关键词: Total graph;    triangular subset;    planarity;    girth;    diameter;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $(x_{1}, . . . , x_{n})$ and $(y_{1}, . . . , y_{n})$ are adjacent if and only if $(x_{1} + y_{1}, . . . , x_{n} + y_{n}) \in U$. 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 $GT^{n}_{U}(U)$, $GT^{n}_U(R^{n}\backslash U)$ and $GT^{n}_{U}(R^{n})$.

【 授权许可】

CC BY