In a simple graph, each pair of vertices can be connected by at most one edge. Therefore, the maximum number of edges is n(n-1)/2.

A graph is acyclic if it contains no cycles. A tree is a connected acyclic graph.

The degree of a vertex is the number of edges that are incident to it.

A bipartite graph is a graph whose vertices can be partitioned into two disjoint sets such that every edge connects a vertex from one set to a vertex from the other set.

The chromatic number of a graph is the minimum number of colors required to color the vertices of the graph such that no two adjacent vertices have the same color.

A Hamiltonian path is a path that visits every vertex in a graph exactly once. A necessary condition for a graph to be Hamiltonian is that it must be connected.

A spanning tree of a connected graph is a subgraph that is a tree and includes all the vertices of the graph. The maximum number of edges in a spanning tree is n-1.

A planar graph is a graph that can be drawn on a plane without any edges crossing each other. A sufficient condition for a graph to be planar is that it must have a vertex of degree at most 3.

A complete bipartite graph is a graph whose vertices can be partitioned into two disjoint sets such that every vertex in one set is connected to every vertex in the other set. The maximum number of edges in a complete bipartite graph with m vertices in one part and n vertices in the other part is mn.

An Eulerian path is a path that visits every edge in a graph exactly once. A necessary condition for a graph to be Eulerian is that it must have an even number of vertices.

In a simple graph, each pair of vertices can be connected by at most one edge. Therefore, the maximum number of edges is m(m-1)/2.

A Hamiltonian path is a path that visits every vertex in a graph exactly once. A necessary condition for a graph to be Hamiltonian is that it must be connected.

A spanning tree of a connected graph is a subgraph that is a tree and includes all the vertices of the graph. The maximum number of edges in a spanning tree is n-1.

A planar graph is a graph that can be drawn on a plane without any edges crossing each other. A sufficient condition for a graph to be planar is that it must have a vertex of degree at most 3.

A complete bipartite graph is a graph whose vertices can be partitioned into two disjoint sets such that every vertex in one set is connected to every vertex in the other set. The maximum number of edges in a complete bipartite graph with m vertices in one part and n vertices in the other part is mn.