边缘分离定理：图的第二小拉普拉斯特征值及其应用
98 浏览量
更新于2024-08-27
收藏 501KB PDF 举报
本文主要探讨了关于图的第二小正则拉普拉斯特征值（second smallest normalized Laplacian eigenvalue）在分割边操作下的行为及其应用。正则拉普拉斯矩阵是图论中的一个关键概念，它在许多领域如网络分析、信号处理和优化问题中扮演着重要角色。第二小特征值λ2(G)对于理解图的结构特性、谱聚类和动力系统稳定性具有重要意义。
研究的焦点在于当图G通过删除或添加一条边进行分割时，λ2(G)如何变化。这种边缘分割操作可能影响图的连通性、局部和全局结构。作者利用理论分析和数值方法，对这一问题进行了深入研究，特别关注了以下几种情况：
1. **树和无环图（Unicyclic graphs）**：在这些图中，λ2(G)通常与树的性质密切相关，因为它们没有环路，所以分割边可能会显著改变λ2(G)。分离边可能导致树变为有向树或减少树的分支程度，从而影响λ2(G)的大小。
2. **连接性变化**：当一条边将图分成两个不相交的部分时，可能会影响图的连通分量数，这直接影响到λ2(G)，因为该特征值与图的连通性紧密相连。如果分割边使得图变得不连通，λ2(G)会受到显著影响。
3. **谱稳定性和复杂度**：在应用谱方法解决实际问题时，如社区检测或网络同步，λ2(G)的变化揭示了图在分割边后结构上的复杂度变化。研究这种行为有助于评估算法的鲁棒性和效率。
4. **应用领域**：结果被广泛应用于各种领域，如无线传感器网络的设计、社交网络分析、生物信息学中的蛋白质结构分析以及机器学习中的图嵌入等。通过了解λ2(G)对分割边的敏感性，可以优化这些领域的算法性能。
该论文通过对一系列理论证明和具体例子的讨论，提供了深入理解第二小正则拉普拉斯特征值在分割操作下的行为框架。这一研究成果不仅增加了我们对图谱理论的理解，而且为实际问题中的优化策略提供了理论基础。这篇论文为研究者和实践者提供了在处理复杂网络时，如何利用第二小特征值来量化和预测分割操作影响的实用工具。
106 J. Li et al. / Discrete Applied Mathematics 171 (2014) 104–115
Fig. 2. Separating an edge uv.
uv (see Fig. 2). In this paper, we study how the second smallest normalized Laplacian eigenvalue behaves when the graph is
perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with λ
2
(G) ≥ 1 −
√
2
2
.
Moreover, the trees and unicyclic graphs with λ
2
(G) = 1 −
√
2
2
are also determined, respectively.
2. An edge-separating theorem on λ
2
(G)
In this section, we consider how the second smallest normalized Laplacian eigenvalue behaves when the graph is
perturbed by separating an edge. Similar problem has been studied for the algebraic connectivity [5]. Before giving our
main result, the following lemmas are needed.
Lemma 2.1 ([8]). Let G = (V, E) be a connected graph and L be its normalized Laplacian matrix. Let f be a harmonic
eigenfunction associated with λ
2
(G). Let v be a cut vertex of G such that G
0
, G
1
, . . . , G
s
are the connected components of the
graph G − v. Then
(1) If f (v) > 0, then exactly one of the components G
i
contains a vertex negatively valuated by f . For all vertices u in the remaining
components f (u) > f (v).
(2) If f (v) = 0 and there exists a component G
i
containing both positively and negatively valuated vertices, then there is exactly
one such component, all remaining components being zero valuated.
(3) If f (v) = 0 and no component contains both positively and negatively valuated vertices, then each component contains either
only positively valuated, or negatively valuated, or zero valuated vertices.
Lemma 2.2 ([6]). Let G be a connected graph with a cutpoint v and α(G) be the algebraic connectivity of G. Then α(G) ≤ 1, the
equality holds if and only if v is adjacent to every vertex of G.
Combined with Proposition 1.3, we have the following:
Lemma 2.3. Let G be a connected graph with a cutpoint v. Then λ
2
(G) ≤ 1. Moreover, if λ
2
(G) = 1 then v is adjacent to every
vertex of G and δ(G) = 1.
Proof. From Proposition 1.3 and Lemma 2.2, we have λ
2
(G) ≤
α(G)
δ(G)
≤
1
δ(G)
≤ 1. Moreover, if λ
2
(G) = 1, then we have
α(G) = 1 and δ(G) = 1. Lemma 2.2 implies that v is adjacent to every vertex of G.
Theorem 2.4. Let e = uv be a cut edge of a connected graph G and suppose that G − uv = G
1
∪ G
2
(|V (G
1
)|, |V (G
2
)| ≥ 2),
where G
1
and G
2
are two components of G − uv, u ∈ V (G
1
) and v ∈ V (G
2
). Let G
′
be the graph obtained from G by separating
the edge uv. Then λ
2
(G) ≤ λ
2
(G
′
), and the inequality is strict if f (v
e
) = 0, where f is a harmonic eigenfunction associated with
λ
2
(G
′
).
Proof. Let V (G
1
) = {u, u
1
, u
2
, . . . , u
n
1
} and V (G
2
) = {v, v
1
, v
2
, . . . , v
n
2
}, f be a harmonic eigenfunction associated with
λ
2
(G
′
). Then V (G
′
) = {u
e
, v
e
, u
1
, u
2
, . . . , u
n
1
, v
1
, v
2
, . . . , v
n
2
}. Let d(x) and d
′
(x) be the degrees of x in G and G
′
, respectively.
Let D
′
and D be the diagonal degree matrices of G
′
and G, respectively. Then d
′
(u
e
) = d(u) + d(v) − 1, d
′
(v
e
) = 1, d
′
(u
i
) =
d(u
i
) for i = 1, . . . , n
1
and d
′
(v
i
) = d(v
i
) for i = 1, . . . , n
2
. From Proposition 1.2, we have λ
2
(G
′
) ≤ 1 since G
′
= K
n
. If
λ
2
(G
′
) = 1, from Lemma 2.3, we have λ
2
(G) < 1 since u (or v) is a cutpoint of G and |V (G
2
)| ≥ 2 (or |V (G
1
)| ≥ 2). Then
the result follows. Now we suppose that λ
2
(G
′
) < 1. Note that −f is also a harmonic eigenfunction associated with λ
2
(G
′
).
We may assume that f (v
e
) ≥ 0. From Proposition 1.1, we have (1 − λ
2
(G
′
))f (v
e
) = f (u
e
). Assume that f (v
e
) = 0. Then
f (u
e
) = 0.
Let h be a vector such that
h(u) = f (u
e
) = 0;
h(v) = f (v
e
) = 0;
h(u
i
) = f (u
i
) for i = 1, 2, . . . , n
1
;
h(v
i
) = f (v
i
) for i = 1, 2, . . . , n
2
.
Then ⟨h, De⟩ =
n
1
i=1
d(u
i
)h(u
i
) +
n
2
i=1
d(v
i
)h(v
i
) =
n
1
i=1
d
′
(u
i
)f (u
i
) +
n
2
i=1
d
′
(v
i
)f (v
i
) = ⟨f , D
′
e⟩ = 0. Moreover,
⟨h, L(G)h⟩ =
xy∈E(G)
(h(x) − h(y))
2
=
xy∈E(G
′
)
(f (x) − f (y))
2
= ⟨f , L(G
′
)f ⟩ and ⟨h, Dh⟩ =
n
1
i=1
d(u
i
)h(u
i
)
2
+
n
2
i=1
d(v
i
)h(v
i
)
2
=
n
1
i=1
d
′
(u
i
)f (u
i
)
2
+
n
2
i=1
d
′
(v
i
)f (v
i
)
2
= ⟨f , D
′
f ⟩.
剩余11页未读，继续阅读
2021-03-30 上传
122 浏览量
2023-08-26 上传
2022-02-05 上传
2020-02-20 上传
2021-05-30 上传
2008-02-27 上传
2021-02-10 上传
2021-02-11 上传
weixin_38605801
- 粉丝: 10
- 资源: 984
我的内容管理
展开
最新资源
- SSM Java项目：StudentInfo 数据管理与可视化分析
- pyedgar：Python库简化EDGAR数据交互与文档下载
- Node.js环境下wfdb文件解码与实时数据处理
- phpcms v2.2企业级网站管理系统发布
- 美团饿了么优惠券推广工具-uniapp源码
- 基于红外传感器的会议室实时占用率测量系统
- DenseNet-201预训练模型：图像分类的深度学习工具箱
- Java实现和弦移调工具：Transposer-java
- phpMyFAQ 2.5.1 Beta多国语言版：技术项目源码共享平台
- Python自动化源码实现便捷自动下单功能
- Android天气预报应用：查看多城市详细天气信息
- PHPTML类：简化HTML页面创建的PHP开源工具
- Biovec在蛋白质分析中的应用：预测、结构和可视化
- EfficientNet-b0深度学习工具箱模型在MATLAB中的应用
- 2024年河北省技能大赛数字化设计开发样题解析
- 笔记本USB加湿器：便携式设计解决方案
