On the genus of non-zero component union graphs of vector spaces
Let $\mathbb{V}$ be an $n$-dimensional vector space over the field $\mathbb{F}$ with a basis $\mathfrak{B}=\{\alpha_1,\ldots,\alpha_n\}.$ For a non-zero vector $v\in\mathbb{V}\setminus\{0\},$ the skeleton of $v$ with respect to the basis $\mathbb{B}$ is defined as $S_\mathfrak{B}(v)=\{\alpha_i : v=\sum_{i=1}^{n} a_i\alpha_i, a_i\neq 0\}.$ The non-zero component union graph $\Gamma(\mathbb{V}_\mathfrak{B})$ of $\mathbb{V}$ with respect to $\mathfrak{B}$ is the simple graph with vertex set $V=\mathbb{V}\setminus\{0\}$ and two distinct non-zero vectors $u,v \in V$ are adjacent if and only if $S_\mathfrak{B}(u)\cup S_\mathfrak{B}(v)=\mathfrak{B}.$ First, we obtain some graph theoretical properties of $\Gamma(\mathbb{V}_\mathfrak{B}).$ Further, we characterize all finite dimensional vector spaces $\mathbb{V}$ for which $\Gamma(\mathbb{V}_\mathfrak{B})$ has genus either 0 or 1 or 2. In the last part of the paper, we characterize all finite dimensional vector spaces $\mathbb{V}$ for which the cross cap of $\Gamma(\mathbb{V}_\mathfrak{B})$ is 1.
___
- [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J.
Algebra, 217, 434–447, 1999.
- [2] I. Beck, Coloring of commutative rings, J. Algebra, 116, 208–226, 1988.
- [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Elsevier, North Hol-
land, Amsterdam, 1986.
- [4] A. Das, Non-zero component union graph of a finite-dimensional vector space, Linear
Multilinear Algebra, 65(6), 1276–1287, 2016.
- [5] A. Das, Nonzero Component graph of a finite dimensional vector space, Comm. Al-
gebra, 44(9), 3918–3926, 2016.
- [6] G. Kalaimurugan, P. Vignesh and T. Tamizh Chelvam, On zero divisor graphs of
commutative rings without identity, J. Algebra Appl. 19(12), # 2050226, 2020.
- [7] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press,
2001.
- [8] K. Selvakumar and P. Subbulakshmi, On the crosscap of the annihilating-ideal graph
of a commutative ring, Palest. J. Math. 7(1), 151–160, 2018.
- [9] M. Syszo, Characterizations of outerplanar graphs, Discrete Math. 26, 47–53, 1979.
- [10] T. Tamizh Chelvam and T. Asir, Genus of total graphs from rings: A survey, AKCE
Int. J. Graphs Comb. 15(1), 97–104, 2018.
- [11] T. Tamizh Chelvam and S. Nithya, Crosscap of the ideal based zero-divisor graph,
Arab J. Math. Sci. 22, 29–37, 2016.
- [12] T. Tamizh Chelvam and K. Prabha Ananthi, On the genus of graphs associated with
vector spaces, J. Algebra Appl. 5, # 2050086, 2019.
- [13] T. Tamizh Chelvam and K. Prabha Ananthi, Complement of the reduced non-
zero component graph of free semi-modules, Accepted for publication in Applied
Mathematics-A Journal of Chinese Universities.
- [14] T. Tamizh Chelvam and K. Selvakumar, On the genus of the annihilator graph of a
commutative ring, Algebra Discrete Math. 24(2), 191–208, 2017.
- [15] A.T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973.