In a graph G with 10 vertices, 7 vertices have degree 3, 2 vertices have degree 4, and 1 vertex has degree 5. Calculate the total number of edges in G.
Sum of degrees = (7 * 3) + (2 * 4) + (1 * 5) = 21 + 8 + 5 = 34. Since 2 * |E| = 34, |E| = 17.
What is the total weight of the Minimum Spanning Tree (MST) for a graph with edges (A,B,2), (B,C,3), (A,C,5), (C,D,1), (B,D,4)?
Edges sorted: (C,D,1), (A,B,2), (B,C,3), (B,D,4), (A,C,5).
Pick (C,D,1)
Pick (A,B,2)
Pick (B,C,3) [Connects A-B-C-D]. Total weight = 1 + 2 + 3 = 6.
What is the minimum number of colors required to properly color the vertices of a cycle graph $C_{7}$ (odd cycle)?
An odd cycle $C_{7}$ requires 3 colors for vertex coloring.
What is the total number of directed edges in a complete directed graph with 6 vertices?
Directed edges in complete graph = n * (n - 1) = 6 * 5 = 30.
What is the maximum possible number of edges in an undirected disconnected simple graph with 7 vertices?
To maximize edges in a disconnected graph with V vertices, isolate 1 vertex and build $K_{V-1}$ on the remaining 6 vertices. Edges = 6 * 5 / 2 = 15.