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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:vector space · finite dimensional · non-zero component graph
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 $S_\mathfrak{B}(v)=\{\alpha_i : v=\sum_{i=1}^{n} a_i\alpha_i, a_i\neq 0\}.$ 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.
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