# How should one define expansion for irregular graphs?

This question is related to a previous question: Does it make sense to talk about expansion in irregular graphs?

A family of expanders can be defined as a sequence of graphs whose spectral gap is bounded from below by a constant. If you want your graphs to be $d$-regular, then the largest spectral gap that you can achieve is $d - 2\sqrt{d-1}$ (this is the Alon-Boppana theorem). These are Ramanujan graphs, for which there are several constructions.

However, many real-world networks are far from regular. Indeed, their degrees tend to obey a power law, so that there are vertices of quite high degree ($n^{\alpha}$ for some $\alpha>0$), even though the graph may have a linear number of edges. If one wants to know whether these graphs are good expanders, it would be nice to compare them to the best expanders with such a degree sequence.

A motivating example for having a new definition for irregular expanders is the "star" graph, in which one vertex is connected to all of the others, and there are no additional edges. Here, although the number of edges is only $n-1$, the spectral gap is enormous - it is $\sqrt{n-1}$. So it looks like the spectral gap can be much larger when the graph is allowed to be non-regular.

So my question is: Is anything known about the maximal expansion among graphs with a given degree sequence?