query: Given a graph $G = (V, E)$, prove $e \leq \frac{n(n-1)}{2}$ for all $n$
positive_passages.docid: 145181
positive_passages.text: How can I prove the maximum number of edges?
negative_passages.docid: 59831
negative_passages.text: How to relate combinatorial proof to bipartite graphs?
negative_passages.docid: 47790
negative_passages.text: If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
negative_passages.docid: 119436
negative_passages.text: Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
negative_passages.docid: 88838
negative_passages.text: How can I prove this inequality for $n\geq 2$?
negative_passages.docid: 44864
negative_passages.text: Prove, formally that: $\log_2 n! \ge n$ , for all integers $n>3$.
negative_passages.docid: 123103
negative_passages.text: Prove recursive formula for number of spanning trees in complete graph.
negative_passages.docid: 76104
negative_passages.text: Bipartite graph. average degree. Euler circuit.
negative_passages.docid: 157898
negative_passages.text: Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
negative_passages.docid: 65096
negative_passages.text: $\chi(G)+\chi(G')\leq n+1$
negative_passages.docid: 145718
negative_passages.text: Proving that A connected graph on $n$ vertices is a tree iff it has $n-1$ edges
negative_passages.docid: 48692
negative_passages.text: A problem about the graph theory
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 87760
negative_passages.text: Determine the diameter of a graph of order $n \geq 3$ where $\delta (G) \geq \frac{n-1}{2}$.
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 91253
negative_passages.text: Proving that every graph that each vertex at degree 2 must contain a cycle
negative_passages.docid: 1453
negative_passages.text: Prove that the graph is connected