On the genus of non-zero component union graphs of vector spaces

Authors : Gnanappirakasam KALAİMURUGAN, Singaravelu GOPİNATH, T TAMİZH CHELVAM

Pages : 1595-1608

Doi:10.15672/hujms.754535

View : 20 | Download : 8

Publication Date : 2021-12-14

Article Type : Research Paper

Abstract :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 $S_\mathfrak{B}insert ignore into journalissuearticles values(v);=\{\alpha_i : v=\sum_{i=1}^{n} a_i\alpha_i, a_i\neq 0\}.$ The non-zero component union graph $\Gammainsert ignore into journalissuearticles values(\mathbb{V}_\mathfrak{B});$ 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 $S_\mathfrak{B}insert ignore into journalissuearticles values(u);\cup S_\mathfrak{B}insert ignore into journalissuearticles values(v);=\mathfrak{B}.$ First, we obtain some graph theoretical properties of $\Gammainsert ignore into journalissuearticles values(\mathbb{V}_\mathfrak{B});.$ Further, we characterize all finite dimensional vector spaces $\mathbb{V}$ for which $\Gammainsert ignore into journalissuearticles values(\mathbb{V}_\mathfrak{B});$ 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 $\Gammainsert ignore into journalissuearticles values(\mathbb{V}_\mathfrak{B});$ is 1.
Keywords : vector space, finite dimensional, non zero component graph, genus