# what is the maximum number of non-loop edges that can exist in an undirected graph

please tell me a equation to find maximum number of non loop edges that can exist in an undirected graph.

for example if vertices are 10 then how many non loop edges can exist?

• Try it with some small numbers. Draw 3, 4, or 5 vertices, then add edges until you can't add any more without making a loop. Then look for a pattern. – MartianInvader Apr 13 '14 at 15:38
• What do you mean by a "loop"? To me, a "loop" is an edge joining a vertex to itself. A graph with two vertices $u,v$ can have as many edges joining $u$ to $v$ as it wants. If you're talking about simple graphs, there can be at most one edge $uv$, but then the question of loops wouldn't even come up. – bof Dec 9 '14 at 15:41
• possible duplicate of How can I prove the maximum number of edges? – Rebecca J. Stones Jun 28 '15 at 4:27

If you have a simple graph, then the extremal case is a complete graph. In which case, there is an edge between each vertex, so there are $\binom{n}{2}$ such edges at most.