Does the correlation coefficient, r, for linear association always exist? On homework assignment I was asked to match different r values (namely $1$, $0.7$, $0.4$, $0$, $-0.4$, $-0.7$, $-1$ and "r value not defined") with some graphs. Among the graphs there were funny looking ones (can be found below) and the "weird" shape, I think, hints at the "not defined" option. 
Are the r values really not defined here? Before this exercise, I was under the impression that every graph in the world can have a linear regression r coefficient—It's just that if the graph is scattered or funny looking, the r value will be about $0$. Is that really the case?

 A: It's possible to have undefined correlation -- what if one of the variables has zero standard deviation?
Consider for example
x   1   1   1   1 
y   0   2   3   5   

That said all those plots in your question have defined correlation... each of them is 0. Indeed, here's a new version of that plot which I generated randomly, with sample correlations (to the printed accuracy):


from comments:

Where in the calculations do we get zero in the numerator..? –    

The numerator has a sum of products $\sum_i (x_i-\bar{x})(y_i-\bar{y})$. Each of those product-terms is a "deviation from horizontal mean, $\bar x$" times "deviation from vertical mean, $\bar y$. That contribution will be positive if both deviations have the same sign (both positive or both negative) and negative if they have opposite signs. Consider the signed area of a rectangle in these images representing contributions of such a product to the sum in the numerator:

(red for positive, blue for negative)
... representing the contributions of two points.
When the picture has left-to-right reflection-symmetry 
then for each region in the plot contributing points like the one shaded red there's a corresponding region the opposite side of $\bar{x}$ contributing points like the one shaded blue (with perfect symmetry there will always be a pair of points that exactly cancel). 
Similarly with top-to-bottom symmetry:

but this time there's a region as far the other side of $\bar{y}$.
As a result, any plot that shows reflective symmetry left to right or top to bottom will have a correlation-numerator of about 0 (or with perfect symmetry, of exactly 0). Every one of those five plots has left-to-right reflective symmetry and the last three have top-to-bottom symmetry as well. Consequently as long as neither variable has variance zero, the correlation will be zero. We can assess these symmetries with a mere glance and immediately and confidently conclude that they indicate no correlation.
A: For some distributions, the correlation coefficient does not exist. For example, the Cauchy distribution. For the estimate of the correlation coefficient, you give me $n>2$ pairs of $(x,y)$. I can estimate it for you given that all of $x$'s are not exact the same AND all of $y$'s are not exact the same.
Suppose in your 5 graphs the x axis are horizontal lines and y axis are vertical lines, I would say their r's are zero.
