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A complete graph is a simple graph in which every two distinct vertices are joined by an edge. The finite complete graph on $n$ vertices is denoted $K_n$. Here the edges are undirected, and self-loops are excluded. The usual pictures below use $n\ge 1$: $K_1$ is a single vertex, $K_2$ a single edge, and $K_3$ a triangle in a graph.^[ref]

Completeness describes all the possible pairs, rather than a drawing style. Different layouts can represent the same graph through a graph isomorphism. Crossing lines away from the drawn vertices do not add new vertices or edges.

There are $\left(\genfrac{}{}{0ex}{}{n}{2}\right)=n(n-1)/2$ unordered pairs of distinct vertices, so this is the edge count of a complete graph. Each vertex has vertex degree $n-1$. The degree-sum formula gives the same count: $2|E|=n(n-1)$. These formulas make a complete graph the upper extreme for edge counts among simple graphs of a fixed size.

This is useful when modelling pairwise relationships. If every pair of objects needs its own direct connection, ten vertices require $45$ edges, while twenty require $190$. Adding one more vertex to $K_n$ introduces $n$ further edges. This counts a specified direct-link model; it does not require every communication system to implement every relationship with a separate physical connection.

A proper vertex coloring of $K_n$ must give every vertex a different color, so its chromatic number is $n$. For $n\ge 1$, its independence number is $1$: two distinct vertices already have an edge between them. Thus complete graphs provide immediate test cases for graph coloring and independent sets.

For a labelled tree with $n\ge 2$ vertices, the vertex degree of label $v$ is one plus the number of occurrences of $v$ in its Prüfer sequence:Each recording of $v$ corresponds to an incident edge removed when a neighbouring leaf of a graph is deleted. Exactly one incident edge remains when $v$ itself is deleted, or when it belongs to the final pair. In particular, $v$ is a leaf of a graph exactly when it never occurs in the Prüfer sequence. The same reasoning applies to each remaining tree and its remaining code. ^[ref]

For $p=(1,3,3,5)$, the occurrence counts in label order are $(1,0,2,0,1,0)$, giving the degree sequence $(2,1,3,1,2,1)$. Its sum is $10=2(6-1)$, as required by the degree-sum formula. More generally, summing the occurrence counts gives $n-2$, so summing the vertex degrees gives $n+(n-2)=2(n-1)$.

A tree is a nonempty connected graph containing no graph cycle. Think of seven locations joined so that everyone can reach everyone else, but no connection supplies an alternative route. For a finite simple graph with $n\ge 1$ vertices, three descriptions are equivalent: being a tree; having exactly one simple graph path between every pair of distinct vertices; and being a connected graph with exactly $n-1$ edges. Here a simple graph path repeats no vertex. Connectedness in the last description is essential. ^[ref]

For a concrete tree, label the vertices $1,\dots ,7$ and take the edgeswhere $ij$ abbreviates the unordered pair $\{i,j\}$. There are seven vertices and six edges. The unique graph path from $4$ to $7$ is $4,2,1,3,6,7$, using five edges. The degree sequence in label order is $(2,3,2,1,1,2,1)$: its sum is $12=2\cdot 6$, as the degree-sum formula predicts. Each of $4,5,7$ is a leaf of a graph, having vertex degree one.

Deleting edge $13$ leaves precisely two connected components of a graph, on $\{1,2,4,5\}$ and $\{3,6,7\}$. Every edge of a tree is a bridge in a graph. Instead add edge $57$ to the original tree: it closes the unique existing graph path $5,2,1,3,6,7$ into exactly one graph cycle. Thus a tree provides economical connections with no redundancy: a single failed link separates locations. Every finite tree with at least two vertices has at least two instances of a leaf of a graph; the endpoints of a longest graph path supply them. ^[ref]

The number $n-1$ alone proves nothing about connectedness: a cycle graph on $1,\dots ,6$ together with an isolated vertex $7$ has seven vertices and six edges, yet is disconnected. Briefly, two different simple graph paths between the same vertices would contain a graph cycle. Repeatedly removing a leaf of a graph and its incident edge proves the tree count $n-1$ by mathematical induction, starting with one vertex and no edges.

For $n\ge 2$, choose $n-2$ independent random variables, each with the uniform distribution on a finite set $\{1,\dots ,n\}$, and decode their values as a Prüfer sequence. Every sequence has probability $n^{-(n-2)}$, and the bijection gives each labelled tree the same probability. This produces a uniform spanning tree of the complete graph without repeatedly proposing sets of edges and rejecting those that fail to be trees. It supplies random inputs for testing implementations of breadth-first search and a tractable model of random connections. ^[ref]

This distribution is uniform over fixed-label trees, rather than over unlabelled trees. With four labels, the star graph shape has probability $4/16=1/4$ and the path graph shape has probability $12/16=3/4$. Also, for a fixed label $v$, the Prüfer degree formula givesHere the binomial distribution counts the independent code positions equal to $v$. The expected value agrees with the average vertex degree from the degree-sum formula. Individual labels have the same distribution by symmetry, though their vertex degrees are constrained together by their fixed sum. When $n=1$, simply return the single isolated vertex.