Consider a weighted complete graph $$G$$ on the vertex set $$\left\{ {{v_1},\,\,\,{v_2},....,\,\,\,{v_n}} \right\}$$ such that the weight of the edge $$\left( {{v_i},\,\,\,\,{v_j}} \right)$$ is $$2\left| {i - j} \right|$$. The weight of a minimum spanning tree of $$G$$ is

If all the edge weights of an undirected graph are positive, then any subject of edges that connects all the vertices and has minimum total weight is a

Let $$G$$ be the simple graph with 20 vertices and 100 edges. The size of the minimum vertex cover of $$G$$ is 8. Then, the size of the maximum independent set of $$G$$ is:

Let $$G$$ be a simple connected planar graph with 13 vertices and 19 edges. Then, the number of faces in the planar embedding of the graph is: