A connected undirected graph has 25 vertices. What is the minimum number of edges it can have?
Every connected graph on V vertices has at least V-1 edges; a tree attains the bound.
A connected undirected graph has 25 vertices. What is the maximum number of edges it can have if it is simple?
Maximum edges are V(V-1)/2 = 25*24/2 = 300.
A tree has 18 vertices. What is the sum of the degrees of all vertices?
Every tree has V-1 edges, and the degree sum is 2E = 2(17) = 34.
A DAG has 9 vertices and no edges. How many topological orderings are possible?
With no precedence constraints, every permutation is valid: 9! = 362880.
$A complete bipartite graph K_{4,7} has how many edges?$
Every vertex in one part connects to every vertex in the other: 4*7 = 28.