1
$\begingroup$

How can I define expander graph with irregular degrees?

Is it possible to define an expander graph with almost regular degrees (all degrees are $d$ or $d-1$.)?

For example, is there any construction algorithm for semiregular graphs?

  • 0
    Are you asking how you can randomly choose an expander graph with those properties? – 2017-01-12
  • 1
    Yes, I want to know that how can I construct an expander graph with irregular or even almost regular degrees? – 2017-01-12
  • 0
    The definition of an expander in this case is the same as in the regular one: You want to have spectral gap bounded from zero in the sequence . As for examples, I would check generalized polygons. There is a book "Spectra of graphs" dealing with spectral calculations. – 2017-01-12
  • 0
    @MoisheCohen Can you explain more? Is there any construction algorithm in this case? – 2017-01-12

1 Answers 1

0

Introduction:

Expander graph is a sparse graph that has strong connectivity properties.

The key idea here is "sparse". In other words, the number edges is relatively small compared to the number of vertices (see this book or here). However the most common definition of expanders is for $d$-regular graphs, that's maybe because one of the following two reasons:

1-Most of the constructed expander graphs are Cayley graphs -which are regular graphs-(some times from the Zig-Zag product).

2-To be able to use Cheeger inequalities (which is also for the d-regular graphs).

Original definition:

The original definition of expanders is stated in Recent Progress In General Topology III (definition 9.2).

Definition 9.2

A finite graph $G$ is a $(k, ε)$-expander if each vertex of $G$ has valency at most $k$, and $h(G) \geq ε > 0$.

A sequence of finite graphs $\{G_i\}$ is called an expander sequence if $|G_i| \rightarrow \infty$ and there exists $k, ε$ such that each $G_i$ is a $(k, ε)$-expander.

Conclusion:

The maximum degree in this definition should be upper bounded. The graph maybe regular or irregular. The graph $G_j$ can be $d$-regular, while the graph $G_{j+1}$ can be $d'$-regular. However, the maximum degree of any graph $G_i$ is less than or equal than some constant $c$ for all $i\in \mathbb{N}^+$.