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$K_{5}$ and $K_{3, 3}$ illustrate complete and complete bipartite graphs, degrees and edge counts

Example

The complete graph $K_{5}$ has degree $4$ at every vertex and has $10$ edges. The complete bipartite graph $K_{3, 3}$ has degree $3$ at every vertex and has $9$ edges.

Facts & Assumptions

Given: The standard graphs $K_{5}$ and $K_{3, 3}$ .

[F1]

$K_{5}$ contains every pair of its five vertices as an edge, while $K_{3, 3}$ has two three-element parts and every edge between the parts (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L1]

$K_{5}$ has $(\binom{5}{2})$ edges (The complete graph on an $n$ -element vertex set has $(\binom{n}{2})$ edges).

[L2]

The sum of degrees is twice the number of edges (Handshake lemma: the sum of the vertex degrees is twice the number of edges).

Verification

technique · direct

1.1F1L1algebra

Each vertex of $K_{5}$ is adjacent to the other four vertices, so its degree is $4$ ; [L1] gives $∣ E (K_{5}) ∣ = (\binom{5}{2}) = 10$ .

1.2F1

Each vertex of $K_{3, 3}$ is adjacent to all three vertices in the opposite part and none in its own part, so all six vertices have degree $3$ .

2.1step 1.2L2algebra∎

By [L2], $2 ∣ E (K_{3, 3}) ∣ = 6 \cdot 3 = 18$ , hence $∣ E (K_{3, 3}) ∣ = 9$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)

K5K5​ and K3,3K3,3​ illustrate complete and complete bipartite graphs, degrees and edge counts