6. If the vertices of a digraph with n vertices can have its vertices numbered from 1 to n such that i > j whenever there is a path from i to j, then such a labeling of its vertices is called sorted labeling. Show that every finite acyclic digraph has a sorted labeling.

How many pairwise non-isomorphic connected simple graphs are there on 6 vertices which contain two vertices of degree 2 and four vertices of degree 3?

please answer

An Euler circuit of the graph K1 is, K1 (z, y, x, W, z, S, W) (V, z, X, W, z, s) (V, Z, x, W, z, s, W) a d. Not possible

6. Prove: “A graph G is a tree if and only if every two vertices of G are connected by a unique path." Provide an illustration to validate your proof.

Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another vertex in the circuit, then there is a circuit in the graph that starts and ends at w.

4. A graph G has 9 vertices and 21 edges. Three vertices have degree x, three have degree y, and three have degree z (where any of the x, y, and z are allowed to be the same number). a) How many graphs G could satisfy these criteria? b) How many graphs G could satisfy these criteria if G has no isolated vertices?