The direct product $G\times H$ has componentwise multiplication. An element $(g,h)$ of finite component orders has order $\mathrm{lcm}(\mathrm{ord}g,\mathrm{ord}h)$.

For a group homomorphism $\varphi :G\to H$, the map $g\ker \varphi \mapsto \varphi (g)$ gives an isomorphism

A group is a set with an associative binary operation, an identity, and an inverse for every element.

If nonnegative quantities satisfythen iteration givesFor $nh\le T$, the growth factor is bounded by $e^{LT}$.

Coordinate compression replaces each fiber parallel to one coordinate axis by an initial interval of the same size. It preserves edges inside fibers and cannot decrease edges between neighboring fibers, because compressed fibers have intersection equal to the smaller fiber size.

For positive $k$, $e^{ik|x-x^{\prime }|}$ is outgoing on both sides of $x^{\prime }$ and obeys

A causal Green function vanishes before the source time and has the derivative jump required by a delta source.

Let $y_1,y_2$ solve $y^{\prime \prime }+\alpha y^{\prime }+\beta y=0$ with $y_1^{\prime }(0)=0$ and $y_2^{\prime }(1)=0$. For Neumann boundary conditions, the Green function iswhere $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$. The derivative jump is one. If $\alpha =0$, the Abel identity makes $W$ constant and $G$ symmetric.

For $\mathrm{Re}m>0$, the decaying solution ofisIt is continuous at zero and has derivative jump $G^{\prime }(0+)-G^{\prime }(0-)=-1$.

König's theorem equates the minimum vertex-cover size of a bipartite graph with its maximum matching size.

The maximum number of disjoint paths joining two vertex sets equals the minimum size of a separating set, with vertex and edge versions according to the chosen notion of disjointness.

The complement of a simple graph has exactly the edges absent from the original graph; a vertex of degree $d$ acquires degree $n-1-d$.

Probabilistic combinatorics uses probability to prove the existence and typical properties of discrete structures.

A random graph is a graph sampled from a probability distribution on graphs.

The Moser spindle is a seven-vertex unit-distance graph with chromatic number four.

A strongly regular graph has constant degree and fixed common-neighbour counts for adjacent and nonadjacent vertex pairs.

If an endpoint of a longest path is adjacent to an internal vertex, replacing the incident path edge by that chord produces another longest path with the same vertex set and a new endpoint. Every neighbour of every endpoint obtained this way must remain on the original path.

Every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ has a Hamilton cycle.

Every sufficiently large graph contains either a prescribed clique or a prescribed independent set.

An antichain is a family of sets no one of which contains another.