How much mathematics are electrical graduate students expected to know? In most Australian universities, the engineering degrees contain three explicit MATH courses which they cover one and two variable calculus, linear algebra, complex numbers, statistics (probability, random variables, etc), Fourier series, differential equations and Laplace transform. For electrical engineering students, they will obviously learn more mathematics but it is usually a part of the electrical engineering subjects and not explicit mathematics subjects. I believe an Australian electrical engineering student does not study topology, group theory, real and functional analysis and other advanced mathematics simply because those topics are not a part of an electrical engineering degree in Australia.

I think (but unsure) that the undergraduate electrical engineering programs in US require students to take much more math subjects.

Let's suppose that someone completes his electrical engineering degree in Australia and he applies for a mathematical research area of electrical engineering (signal processing, control theory) in graduate school in USA. Suppose that he is a good candidate overall; will graduate admission committee be concerned about Australian students mathematical ability? How do electrical engineering graduate school make sure that the Australian applicants have the required mathematical skills to complete the graduate program successfully?
I think it really depends. I've seen EE majors do much of the math someone majoring in math. At the junior-senior level, some EE students will take a semester of advanced calculus, matrix theory, and numerical analysis. It really depends on the university and their degree requirements.

For students outside of the US, admissions committees do try to review the student's background as carefully as possible, comparing it to their requirements. Certainly it will help to have as much math as possible if you want to stand out. In particular, try taking some "applied" courses - partial differential equations, numerical analysis, optimization, stuff like that. Some "pure" topics like real analysis (perhaps up to introductory measure theory, this is really not a big thing in my opinion) and complex analysis will help too.

If you plan to apply to a US university, I suggest talking to someone in graduate admissions in the department that you are interested in.