Draw the complete bipartite graph K4, 4. Label its vertices 1, 2, 3, ., n and list the edges in lexicographic order. ) Use the labeling of the vertices (a) to write the adjacency matrix of the graph K4, 4.

Let G be a polyhedral graph with 12 vertices and 30 edges. Prove that the degree of each face of G is three. Show all of your work and clearly explain, using words, your reasoning.

Graph Complement and Planarity,(a) Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph.  for this problem, you must writecomplete sentences, include all details, show all of your work, and clarify all of your reasoning.

Identify and write Euler circuit and Euler path. Also mention vertices, edges and degree of all vertices of the following graph. f A d D a B