This is a long and detailed hint which provides ideas, not just for this particular problem, but also for other magic graph problems where similar reasoning can be used.

Vertex Magic Investigation
There are 8 vertices and 7 edges. Let's suppose the magic constant is $h$ and this is the same at each vertex. The total of all the numbers 1 + 2 + ... + 15 = 120 but the numbers on the edges, say $a, b, c, d, e, f, g$ are each counted twice so

$(a + b + c + d + e + f + g) + 120 = 8 h .$

So $a + b + c + d + e + f + g$ is a multiple of 8 and it is at least $1 + 2 + . . . + 7 = 28$ so it must be 32 or more. Hence $8 h \geq 152$ and $h \geq 19$ .

It is easy to check, in a similar way, for the largest value that $h$ can take. Now it is a matter of systematically checking the possibilities for magic labellings with $h = 19$ etc.

The techniques for finding the labellings are the same as used for the Caterpillar problem.

Edge Magic Investigation
How do we discover what edge magic graphs there are? Here the three order-3 vertices

a, b, c

are counted three times and the sum of all the numbers is again 120.

2 (a + b + c) + 120 = 7 k .

Hence

k

is even and since

a + b + c \geq 6

we have

7 k \geq 120 + 12 = 132

and so

k \geq 20

If $k = 20$ then $a + b + c = 10$ so this triple is 1+2+7 or 1+3+6 or 1+4+5 or 2+3+5.

The possible edge sums for $k = 20$ are:
1+4+15, 1+5+14, 1+6+13, 1+7+12, 1+8+11, 1+9+10, 2+3+15, 2+4+14, 2+5+13, 2+6+12, 2+7+11, 2+8+10, 3+4+13, 3+5+12, 3+6+11, 3+7+10, 3+8+9, 4+5+11, 4+6+10, 4+7+9, 5+6+9, 5+7+8

It is easy to find the edge magic labelling for k=20.

Moreover, substituting $16 - P$ for each of the P-labels giving a magic total of k will give the same numbers 1 to 15 and a magic total of 48-k. So for all the magic graphs you find with a magic total of 20 there are corresponding graphs with a magic totl of 28.

It is left for you to find the edge magic labels for $k = 20$ and for $k = 22$ (and correspondingly $k = 26$ ) andfor $k = 24$ . Remember $k$ must be even and at least 20 so these are the only possibilities.