Let <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> be a graph with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$V(G)=\{v_1,v_2,\ldots ,v_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overrightarrow{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> be the associated digraph of G by replacing each of its edges by two oppositely oriented arcs with the same ends. A complex unit gain graph, denoted by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi =(G,\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is a triple <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\overrightarrow{G},\mathbb {T},\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">→</mo> </mover> <mo>,</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> consisting of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\overrightarrow{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>, the circle group <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation> and a gain function <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varphi : {E}(\overrightarrow{G})\rightarrow \mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">T</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varphi (\overrightarrow{e_{ij}})=\varphi (\overrightarrow{e_{ji}})^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <msub> <mi>e</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>φ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <msub> <mi>e</mi> <mrow> <mi mathvariant="italic">ji</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$e_{ij}=v_iv_j\in E(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>e</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>∈</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {T}=\{z\in \mathbb {C}:|z|=1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we introduce the Laplacian energy of a complex unit gain graph. We prove that the Laplacian energy of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$-\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation> is invariant under switching equivalence of complex unit gain graphs, and then present some bounds for the Laplacian energy of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$-\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation> in terms of the first Zagreb index of G and the real parts of the gains of the oriented cycles in suitably oriented graph of G, respectively.

Bounds for the Laplacian energy of complex unit gain graphs

期刊论文
发表日期：

2026年7月10日

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Jing Zhao,
Huiqing Liu

摘要

Let $G=(V(G),E(G))$ $G = (V (G), E (G))$ be a graph with $V(G)=\{v_1,v_2,\ldots ,v_n\}$ $V (G) = {v_{1}, v_{2}, \dots, v_{n}}$ , and let $\overrightarrow{G}$ $\vec{G}$ be the associated digraph of G by replacing each of its edges by two oppositely oriented arcs with the same ends. A complex unit gain graph, denoted by $\Phi =(G,\varphi )$ $Φ = (G, φ)$ , is a triple $(\overrightarrow{G},\mathbb {T},\varphi )$ $(\vec{G}, T, φ)$ consisting of $\overrightarrow{G}$ $\vec{G}$ , the circle group $\mathbb {T}$ $T$ and a gain function $\varphi : {E}(\overrightarrow{G})\rightarrow \mathbb {T}$ $φ : E (\vec{G}) \to T$ such that $\varphi (\overrightarrow{e_{ij}})=\varphi (\overrightarrow{e_{ji}})^{-1}$ $φ (\vec{e_{ij}}) = φ {(\vec{e_{ji}})}^{- 1}$ for $e_{ij}=v_iv_j\in E(G)$ $e_{ij} = v_{i} v_{j} \in E (G)$ , where $\mathbb {T}=\{z\in \mathbb {C}:|z|=1\}$ $T = {z \in C : | z | = 1}$ . In this paper, we introduce the Laplacian energy of a complex unit gain graph. We prove that the Laplacian energy of $-\Phi $ $- Φ$ is invariant under switching equivalence of complex unit gain graphs, and then present some bounds for the Laplacian energy of $-\Phi $ $- Φ$ in terms of the first Zagreb index of G and the real parts of the gains of the oriented cycles in suitably oriented graph of G, respectively.