Vertex Expansion in Graph Theory

• Vertex expansion is a graph-theoretic measure that quantifies how new vertices are reached across a set's boundary, with variants such as classical, symmetric, and robust forms.
• It plays a crucial role in graph partitioning, small-set expansion, and rumor spreading, linking combinatorial properties with spectral quantities like Laplacian eigenvalues.
• Recent advances explore approximation algorithms, hardness results, and spectral relaxations, extending vertex expansion to hypergraph reductions, clustering, and linear-algebraic analogues.

to=arxiv_search.search 大发快三计划 天天彩票软件_json string={"query":"vertex expansion graph Cheeger small-set vertex expansion", "max_results": 10, "sort_by": "relevance"} 丁香五月 to=arxiv_search.search code 天天送彩票json {"query":"vertex expansion graph Cheeger small-set vertex expansion","max_results":10,"sort_by":"relevance"} Let's wait. to=arxiv_search.search 񎔄json ุ้นบาท{"query":"vertex expansion graph Cheeger small-set vertex expansion","max_results":10,"sort_by":"relevance"} to=arxiv_search.search 在天天中彩票json 北京赛车女郎{"query":"all:vertex expansion graph", "max_results": 10, "sort_by": "submittedDate"} Vertex expansion is a graph-theoretic isoperimetric parameter that measures how rapidly a set acquires new vertices across its boundary. In its classical form, for a $d$-regular graph $G=([n],E)$, the outer vertex boundary is

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$

and the vertex expansion is

$\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$

Other standard normalizations use symmetric cut ratios, weighted boundaries, or robust boundary size. Across these formulations, vertex expansion is a central notion in graph partitioning, small-set expansion, Cheeger-type inequalities, rumor spreading, and several linear-algebraic analogues (Li et al., 2022, Louis et al., 2013, Kwok et al., 2022).

1. Classical definitions and normalizations

The notion is not unique up to notation, but the standard variants are tightly related. One family counts new vertices reached from a set, another symmetrizes the cut, and a third incorporates vertex weights. A further refinement, robust vertex expansion, measures how many boundary vertices are needed to capture a constant fraction of the outgoing edge mass.

Variant	Formula	Typical setting
Classical outer-boundary form	$$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$	$d$-regular graphs
Symmetric cut normalization	$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$	approximation/hardness
Weighted form	$\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$	reversible chains, weighted graphs
Robust form	$\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$	Cheeger refinements

The symmetric normalization

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$

is accompanied by the symmetric vertex expansion

$G=([n],E)$0

and these forms are computationally equivalent up to constant factors via explicit reductions (Louis et al., 2013). In the weighted setting,

$G=([n],E)$1

where the outer $G=([n],E)$2 keeps the parameter comparable with spectral quantities bounded by $G=([n],E)$3 (Kwok et al., 2022).

Robust vertex expansion is defined by

$G=([n],E)$4

It records how broadly the outgoing edge mass of $G=([n],E)$5 is distributed over boundary vertices, rather than merely counting whether a boundary vertex is present (Kwok et al., 2015).

2. Relations to edge expansion and spectral quantities

For bounded-degree graphs, vertex expansion, edge expansion, and spectral expansion are equivalent up to constants. If $G=([n],E)$6 is $G=([n],E)$7-regular, with normalized edge expansion $G=([n],E)$8 and normalized Laplacian second eigenvalue $G=([n],E)$9, then

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$0

Hence a bounded-degree family is an expander in any one of the three senses if and only if it is an expander in the others (Li et al., 2022).

Vertex expansion also admits spectral surrogates that are distinct from the ordinary Laplacian gap. One such parameter is

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$1

which satisfies

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$2

This parameter behaves as a vertex-expansion analogue of a spectral gap, but direct optimization is NP-hard, which motivates SDP relaxations and reweighted spectral theories (Louis et al., 2013).

The relation to edge conductance is subtler outside the regular setting. Robust vertex expansion enters sharpened Cheeger inequalities through the product

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$3

which interpolates between ordinary edge expansion and the spread of the boundary over vertices. This refinement is central when outgoing edges concentrate on few vertices, a regime in which edge and vertex notions diverge sharply (Kwok et al., 2015).

3. Approximation algorithms and hardness

The current worst-case approximation landscape is anchored by a sharp Small-Set-Expansion-based hardness result and matching upper bounds up to constants and logarithmic factors. For a graph of maximum degree $$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$4, there is a polynomial-time algorithm that outputs a set $$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$5 with

$$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$6

Under the Small Set Expansion hypothesis, this is asymptotically tight: it is hard to find a subset with expansion less than $$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$7, and in particular it is SSE-hard to distinguish whether the vertex expansion is $$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$8 or at least an absolute constant. The same work introduces Analytic Vertex Expansion as an intermediate problem and uses Gaussian isoperimetry and an invariance principle to transfer hardness from SSE to vertex expansion (Louis et al., 2013).

This degree dependence distinguishes vertex expansion from edge expansion. The analogous threshold for edge expansion is $$\partial_{\mathrm{out}}(W)=\{j\in [n]\setminus W:\exists i\in W\ \text{s.t.}\ \{i,j\}\in E\},$$9 with no dependence on the degree, whereas the vertex-expansion threshold carries a $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$0 factor (Louis et al., 2013). A plausible implication is that vertex expansion is inherently more sensitive to local degree structure than edge conductance.

For $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$1-Small-Set Vertex Expansion, recent progress uses stronger convex relaxations. A randomized algorithm running in time

$\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$2

outputs a set $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$3 of size $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$4 with vertex expansion at most

$\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$5

where $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$6 is the largest vertex degree and $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$7 is the optimal $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$8-SSVE. The method uses the basic SDP relaxation augmented with $\mu(G)=\min_{\substack{W\subseteq [n]\1\le |W|\le n/2}}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}.$9 rounds of the Lasserre/SoS hierarchy, a rounding algorithm combining Raghavendra–Tan and Austrin–Benabbas–Georgiou, and a Gaussian rounding lemma for hyperedges (Ghoshal et al., 2023).

4. Small sets, hypergraph reductions, and $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$0-way variants

Small-set vertex expansion is the regime

$$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$1

and it is algorithmically tied to hypergraph expansion. A reduction constructs a hypergraph $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$2 from a graph $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$3 by assigning to each vertex $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$4 a hyperedge

$$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$5

so that small-set vertex expansion becomes equivalent, up to constants, to hypergraph small-set expansion. This yields a randomized polynomial-time $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$6 approximation algorithm for SSVE and an instance-sensitive guarantee that finds a set with vertex expansion

$$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$7

from the corresponding hypergraph algorithm (Louis et al., 2014).

The $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$8-way problem replaces a single cut by a partition $$\mu(G)=\min_{1\le |W|\le n/2}\frac{|\partial_{\mathrm{out}}(W)|}{|W|}$$9 and optimizes

$d$0

For vertex expansion the paper on planted models uses the symmetric normalization

$d$1

and studies balanced $d$2-partitions with $d$3. In the planted model $d$4, if

$d$5

there is a polynomial-time algorithm returning disjoint sets $d$6 such that

$d$7

and then a $d$8-partition $d$9 with

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$0

(Louis et al., 2019).

These formulations show that vertex expansion is not confined to balanced two-way cuts. It supports a small-set regime tied to SSE and Strong Unique Games (Ghoshal et al., 2023), a hypergraph interface (Louis et al., 2014), and planted $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$1-partition models that are closer to clustering and community detection (Louis et al., 2019).

5. Robust and reweighted spectral theories

Two distinct spectral refinements have emerged for vertex expansion. The first uses robust vertex expansion and the classical Laplacian. Defining

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$2

one obtains the generalized Cheeger bounds

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$3

These inequalities explain why spectral partitioning improves when robust vertex expansion is large or when there are few disjoint non-expanding sets, and they extend to local partitioning algorithms based on personalized PageRank and truncated random walks (Kwok et al., 2015).

The second refinement replaces the ordinary spectral gap by a maximum reweighted spectral gap. For a target stationary distribution $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$4, one considers all reversible Markov chains on $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$5 with stationary distribution $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$6, and defines $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$7 as the maximum reweighted second smallest eigenvalue. In this framework,

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$8

where $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$9 is the maximum degree. The same reweighted theory extends to weighted vertex expansion, gives an analogue of Trevisan’s bipartiteness result, analogues of higher order Cheeger and improved Cheeger inequalities, and produces $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$0-polytopes whose graphs have very poor vertex expansion (Kwok et al., 2022).

These two spectral theories address different structural questions. Robust vertex expansion sharpens Laplacian-based guarantees by tracking how boundary mass is distributed across vertices, while reweighted eigenvalues characterize how much spectral gap can be created by changing edge weights subject to reversibility and support constraints. Together they show that vertex expansion admits a spectral theory substantially richer than the classical $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$1-conductance correspondence.

6. Examples, dynamics, and linear-algebraic analogues

High girth does not by itself force lossless vertex expansion on all sublinear scales. For every $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$2 for prime $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$3 and infinitely many $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$4, there exists an $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$5-vertex $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$6-regular graph with girth $\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$7, nontrivial eigenvalues bounded in magnitude by

$\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$8

and vertex expansion of sublinear sized sets bounded by

$\psi(S)=\frac{\pi(\partial S)}{\pi(S)}$9

At the same time, in any Ramanujan graph with girth $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$0, all sets of size bounded by $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$1 have vertex expansion $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$2 (McKenzie et al., 2020).

Vertex expansion also controls information diffusion. For the PUSH-PULL rumor spreading protocol, if a connected graph has vertex expansion at least $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$3, then all $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$4 nodes are informed in

$\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$5

rounds with high probability, where $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$6 is the maximum degree. This matches the known lower bound and establishes a tight dependence of rumor-spreading time on vertex expansion (Giakkoupis, 2013).

A linear-algebraic analogue is dimension expansion. For a tuple $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$7 and a subspace $\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$8,

$\phi^V(S)=\frac{N_{1/2}(S)}{|S|}$9

plays the role of graph vertex expansion. It is equivalent, up to a factor $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$0, to dimension edge expansion $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$1, but it is strictly weaker than quantum expansion. For graphical tuples $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$2, the identifications

$$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$3

make the analogy exact (Li et al., 2022).

7. Terminological extensions beyond graph theory

Outside graph theory, the phrase vertex expansion is used in unrelated technical senses. In constructive quantum field theory, a vertex expansion reorganizes perturbative series so that the basic building blocks are effective vertices, often loop vertices, connected by trees or forests rather than by arbitrary Feynman graphs. In the Loop Vertex Expansion for $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$4 theories, the key ingredient is “forcing the integration of exactly one particular field per vertex of the initial action,” which yields a loop vertex action with controlled derivatives and a convergent tree expansion (Rivasseau, 2017).

In scattering amplitudes, the MHV and super MHV vertex expansions express tree-level amplitudes in $$\Phi^V_G(S)=n\cdot \frac{|N_G(S)|}{|S||V\setminus S|}$$5 SYM as sums of diagrams built from on-shell MHV vertices connected by scalar propagators. The supersymmetric version depends on a reference spinor and four reference Grassmann parameters, and a judicious choice of those parameters makes a significant fraction of diagrams vanish (0903.0377).

These usages are terminologically parallel rather than conceptually continuous with graph-theoretic vertex expansion. In graph theory the term is isoperimetric and combinatorial; in constructive field theory and amplitude theory it denotes a reorganization of perturbative objects around effective vertices.