---
title: "Cheeger bound"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Cheeger_bound"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:21:47.049744+00:00"
instance: "kb-cron"
---

In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs.
Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a finite set and let 
  
    
      
        K
        (
        x
        ,
        y
        )
      
    
    {\displaystyle K(x,y)}
  
 be the transition probability for a reversible Markov chain on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. Assume this chain has stationary distribution 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
.
Define

  
    
      
        Q
        (
        x
        ,
        y
        )
        =
        π
        (
        x
        )
        K
        (
        x
        ,
        y
        )
      
    
    {\displaystyle Q(x,y)=\pi (x)K(x,y)}
  

and for 
  
    
      
        A
        ,
        B
        ⊂
        X
      
    
    {\displaystyle A,B\subset X}
  
 define

  
    
      
        Q
        (
        A
        ×
        B
        )
        =
        
          ∑
          
            x
            ∈
            A
            ,
            y
            ∈
            B
          
        
        Q
        (
        x
        ,
        y
        )
        .
      
    
    {\displaystyle Q(A\times B)=\sum _{x\in A,y\in B}Q(x,y).}
  

Define the constant 
  
    
      
        Φ
      
    
    {\displaystyle \Phi }
  
 as

  
    
      
        Φ
        =
        
          min
          
            S
            ⊂
            X
            ,
            π
            (
            S
            )
            ≤
            
              
                1
                2
              
            
          
        
        
          
            
              Q
              (
              S
              ×
              
                S
                
                  c
                
              
              )
            
            
              π
              (
              S
              )
            
          
        
        .
      
    
    {\displaystyle \Phi =\min _{S\subset X,\pi (S)\leq {\frac {1}{2}}}{\frac {Q(S\times S^{c})}{\pi (S)}}.}
  

The operator 
  
    
      
        K
        ,
      
    
    {\displaystyle K,}
  
 acting on the space of functions from 
  
    
      
        
          |
        
        X
        
          |
        
      
    
    {\displaystyle |X|}
  
 to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
, defined by

  
    
      
        (
        K
        ϕ
        )
        (
        x
        )
        =
        
          ∑
          
            y
          
        
        K
        (
        x
        ,
        y
        )
        ϕ
        (
        y
        )
      
    
    {\displaystyle (K\phi )(x)=\sum _{y}K(x,y)\phi (y)}
  

has eigenvalues 
  
    
      
        
          λ
          
            1
          
        
        ≥
        
          λ
          
            2
          
        
        ≥
        ⋯
        ≥
        
          λ
          
            n
          
        
      
    
    {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}}
  
.  It is known that  
  
    
      
        
          λ
          
            1
          
        
        =
        1
      
    
    {\displaystyle \lambda _{1}=1}
  
.   The Cheeger bound is a bound on the second largest eigenvalue 
  
    
      
        
          λ
          
            2
          
        
      
    
    {\displaystyle \lambda _{2}}
  
.
 Theorem (Cheeger bound):

  
    
      
        1
        −
        2
        Φ
        ≤
        
          λ
          
            2
          
        
        ≤
        1
        −
        
          
            
              Φ
              
                2
              
            
            2
          
        
        .
      
    
    {\displaystyle 1-2\Phi \leq \lambda _{2}\leq 1-{\frac {\Phi ^{2}}{2}}.}
  


== See also ==
Stochastic matrix
Cheeger constant
Conductance


== References ==