A simple graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G=(V(G),E(G))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is considered as a k-dot product graph when there exists a function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f:V(G)\longrightarrow \mathbb {R}^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟶</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(u).f(v)\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for any two adjacent vertices u and v. The dot product dimension <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\rho (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of G is the minimum value k for which G is a k-dot product graph. In this paper, we provide a characterization of disconnected graphs with exactly one cycle with respect to their dot product dimension.

A Classification of Disconnected Graphs with Exactly One Cycle by Their Dot Product Dimension

期刊论文
发表日期：

2026年2月6日

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Mahin Bahrami,
Dariush Kiani

摘要

A simple graph $G=(V(G),E(G))$ $G = (V (G), E (G))$ is considered as a k-dot product graph when there exists a function $f:V(G)\longrightarrow \mathbb {R}^k$ $f : V (G) ⟶ R^{k}$ such that $f(u).f(v)\ge 1$ $f (u) . f (v) \geq 1$ for any two adjacent vertices u and v. The dot product dimension $\rho (G)$ $ρ (G)$ of G is the minimum value k for which G is a k-dot product graph. In this paper, we provide a characterization of disconnected graphs with exactly one cycle with respect to their dot product dimension.