7. 8. 9. 10. Let G be a regular graph with at least one edge. Assume that there is a way of partitioning the set of vertices of G into two sets X, Y such that this partition makes G bipartite. Prove that |X| = |Y. Let G be a regular graph with a vertex of degree 4 and 10 edges. How many vertices does it have? Prove that the complement of a regular graph is also regular. Prove that a bipartite graph on n vertices has a number of edges less or equal to 2

I want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.

2.12 Prove that a 3-regular graph has a cut vertex if, and only if, it has some bridge.

Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.