Non-regular Expander graph ?! The answer to the following question could be **trivial**. Expander graph is a sparse graph that has strong connectivity properties. In "Expander Families and Cayley Graphs" Book or here you will find the following definition of expander graph:  will be sequence of graph where $\forall n$ we have $\Delta(X_n)\leq c$, for some $c\in \mathbb{N}^+$ ( Note that $\Delta(X_n)$ is the max degree of graph $(X_n)$). Any help will be useful!

At last I have reached an answer for my question.

And the answer is yes! We can define expander family for the non-regular case.

The most common definition for 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)

2-To be able to use Cheeger inequalities.

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.