In mathematical exercises, 'straight line' is redundant. All lines are defined as straight. The fact that endpoints are given (from the center, to the center), further defines them as segments and not infinite lines.
Let's see. Taking the upper left circle to start with, the line from that one to the four circles on the adjacent branches, as well as the one touching it would not intersect any other centers. Likewise, the ones going to the 'roots' of the non-opposite branches would not intersect any centers. The one to the center circle would go through one, as would the two that go to the tips of the non-opposite branches. The one going to the 'root' of the opposite branch would go through two centers, and the one going to the tip of the opposite branch would go through three, so those don't count.
So, for each outer circle, there would be three lines that go through one other circle's center.
Each inner circle would have exactly one line that goes through one other circle's center (namely, the one opposite the center circle).
The center circle would have six lines (from the center to the tip of each branch) that go through one other circle's center.
So, 6+6+18 = 30 - but, each of those segments is actually counted twice (once from each end), so my final answer is 15.