Can we have an equation for any graph?

Question

I was wondering, can we have an equation for any graph even the most zigzagged one?

Answer 1

There are many, many more graphs in the plane than there are formulas we can write down. In fact, since equations must be written in finitely many characters from a finite alphabet, the number of possible equations - or definite descriptions of any kind - we can write is a countable infinity. However, there are uncountably many functions with different graphs on the plane.

I would have a hard time, of course, describing to you in a definite way one of the uncountable multitude of graphs for which there exists no definite description. They're everywhere, though.

Answer 2

I would say the apt way of representing graphs are not equations, but functions (a function can reduce to an equation in x for 2d, if the function is smooth and continuous, by using some sort of spline interpolation or using nth order polynomial fit). On the other hand any curves in 2d plane atleast, you can write for the worst case, as discontinuous functions representing the graph, even if the function has a limit going to infinity at some point. But one may not want to do this if the graph is too complicated.

Answer 3

If you specify any finite collection of points with distinct $x$ co-ordinates and say you want a graph that will go through them, there is an equation, a polynomial even, that will go through all your points. This is called an interpolating polynomial and is not too hard to construct. I wrote an article on interpolating polynomials that explains the process a bit.

If you draw a line on a page with a pencil (so the line has nonzero thickness), there is even a polynomial that stays within that pencil line, as long as it's reasonable (no gaps in the line, no going back on yourself, etc.). That is to say, any continuous graph can be approximated as close as you like (e.g. within a pencil stroke's thickness) with polynomials; one method of doing so is using Bernstein polynomials.

Hence, I suspect the answer to the question you were trying to ask is "yes".

But that's sort of besides the point. The important thing to understand is that "equations" can really be whatever you like. I can write an equation, if it pleases me, of the following form:

\[y = \text{the vertical distance of Graph A from the $x$-axis, at the point distance $x$ from the $y$-axis}\]

As long as I also show you what Graph A is, this is a statement of equality between two well-defined numeric expressions: an equation. So by definition, there is an equation associated with every graph.

You may be more interested in questions like "suppose I have a graph with property $P$, can I match it with an equation with property $Q$". The answers to these questions of course depend on what $P$ and $Q$ are, but as shown in the beginning of my answer, there are a lot of good results you can find.