In graph theory, a path is a sequence of vertices such that consecutive vertices are connected by edges. What is the maximum number of edges in a path with n vertices?

A cycle in a graph is a path that begins and ends at the same vertex. What is the minimum number of edges in a cycle with n vertices?

A graph is called connected if there is a path between every pair of vertices. What is the maximum number of edges that can be removed from a connected graph with n vertices so that it remains connected?

A tree is a connected graph with no cycles. What is the maximum number of edges that a tree with n vertices can have?

A graph is called Eulerian if it contains a cycle that visits every edge exactly once. What is a necessary and sufficient condition for a graph to be Eulerian?

A graph is called Hamiltonian if it contains a cycle that visits every vertex exactly once. What is a sufficient condition for a graph to be Hamiltonian?

The girth of a graph is the length of the shortest cycle in the graph. What is the girth of a tree?

The chromatic number of a graph is the minimum number of colors needed to color the vertices of the graph so that no two adjacent vertices have the same color. What is the chromatic number of a complete graph with n vertices?

The independence number of a graph is the maximum number of vertices that can be selected such that no two selected vertices are adjacent. What is the independence number of a complete graph with n vertices?

The clique number of a graph is the maximum number of vertices that can be selected such that every pair of selected vertices is adjacent. What is the clique number of a complete graph with n vertices?

The domination number of a graph is the minimum number of vertices that need to be selected such that every other vertex is adjacent to at least one selected vertex. What is the domination number of a complete graph with n vertices?

The matching number of a graph is the maximum number of edges that can be selected such that no two selected edges share a common vertex. What is the matching number of a complete graph with n vertices?

The vertex cover number of a graph is the minimum number of vertices that need to be selected such that every edge of the graph is incident to at least one selected vertex. What is the vertex cover number of a complete graph with n vertices?

The edge cover number of a graph is the minimum number of edges that need to be selected such that every vertex of the graph is incident to at least one selected edge. What is the edge cover number of a complete graph with n vertices?

The arboricity of a graph is the minimum number of forests into which the edges of the graph can be partitioned. What is the arboricity of a complete graph with n vertices?