Let $G$ be a graph of order $n$ with $\kappa (G) \geq 1$. Prove that $n> \kappa(G)[\operatorname{diam}(G)-1]+2$
102539
Min. number of vertices in graph as function of $\kappa(G)$ and $\operatorname{diam}(G)$
28266
For $n\geq 3$ there exist $x,y\in S_n$ such that $x$ and $y$ have order $2$ and $xy$ has order $n$.
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
107716
Show that $G$ is cyclic
82935
Prove that f∈Θ(g)if and only ifO(f) =O(g).
52261
Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$
121264
Prove that rank $(g \circ f)\le \min[\operatorname{rank}(f),\operatorname{rank}(g)]$; $g$ and $f$ are linear mappings
133270
Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
7418
The Center of $\operatorname{GL}(n,k)$
2455
Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
104855
Prove that $N \subset Z(G)$
87760
Determine the diameter of a graph of order $n \geq 3$ where $\delta (G) \geq \frac{n-1}{2}$.
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
50525
Prove that $(n, n, n-1, n-1, \ldots , 4, 4, 3, 3, 2, 2, 1, 1)$ is a graphic!