Question

4. Find the generating function for the number of labeled graphs where there are 1 or 2 edges at each vertex.

Answer 1

ANSWER :-

Given that 1 or 2 edges at each vertex.

Here we need to find the generating function for the number of labeled graphs :

G(x) is expected to be the exponential producing capacity for the number of connected graphs on n labelled vertices with greatest edges 2

where G(x) is a generating function

There are two sorts of diagrams this way: ways and cycles.

There are $\frac{n!}{2}$ paths, provided $n\geq 2$. We can name the vertices on a way in n! ways, however turning around the request delivers a similar way.

There are (n−1)!2 cycles, . Erasing vertex n produces a way on (n−1)! vertices,

Therefore the generating function.

0 0 2 G(x) = n 2 n! n2 2

the number of graphs of this structure with an odd number of associated parts

Thus we find a generating function for the no.of labelled graphs where there are 1 or 2 edges at each vertex.

Thank you