The data structure linked list allows another^{1} representation for digraphs or a graphs to be used in computers, which is convenient in cases where the number of edges is not too big (e.g. less then the square of the number of vertices \(E < V^2\)).
Let \(D=(V,E,\alpha,\omega)\) be a digraph and let \(L\) be the set of all linked lists of vertices in \(V\), i.e.
\[L:=\{(n_{i_1},n_{i_2},\ldots),~n_{i_j}\in V\}\]
An adjacency list of \(D\) is a function. \[ f:=\cases{ V \to L \\ n_i \to (n_{i_1},n_{i_2},\ldots)\Longleftrightarrow n_{i_j}\in N^+(n_i), } \]
i.e. \(f\) assigns to each vertex \(n_i\) a list of all of its adjacent successors \(n_{i_j}\), i.e. the edges \(n_in_{i_j}\) exist and the vertex \(n_i\) is their initial vertex, while the vertices \(n_{i_j}\) are their terminal vertices.
The adjacency list for an undirected graph \(G=(V,E,\gamma)\) is defined correspondingly:
\[ f:=\cases{ V \to L \\ n_i \to (n_{i_1},n_{i_2},\ldots)\Longleftrightarrow n_{i_j}\in N(n_i), } \]
i.e. \(f\) assigns to each vertex \(n_i\) a list of all of its neighbors \(n_{i_j}\), i.e. the edges \(n_in_{i_j}\) exist and the vertices \(n_i\) and \(n_{i_j}\) are adjacent.
Digraph  Adjacency list 

A digraph \(D\) with \(6\) vertices and some edges.  `[ 
\begin{array}{ccl}
a & \rightarrow & (b,c,d)\cr
b & \rightarrow & (a,a) \cr
c & \rightarrow & (b,b,b) \cr
d & \rightarrow & (e,e) \cr
e & \rightarrow & (e) \cr
f & \rightarrow & () \cr
\end{array}
]Please note that in an adjacency list representation of a digraph
* the order of the successors in each list is irrelevant, all that counts is the number of occurrences of each successor,
* if a vertex is in its own list, this indicates a loop,
* multiple occurrences of nodes in the list indicate multiple edges,
* empty lists
(())indicate isolated nodes (like for the node
(f)). 
Graph Adjacency list

![graphs6](https://github.com/bookofproofs/bookofproofs.github.io/blob/main/_sources/_assets/images/examples/graphs6.jpg?raw=true) A graph
(G)with
(6)vertices and some edges.
[
\begin{array}{ccl}
a & \rightarrow & (b,b,b,c,d)\cr
b & \rightarrow & (a,a,a,c,c,c) \cr
c & \rightarrow & (a,b,b,b) \cr
d & \rightarrow & (a,e,e) \cr
e & \rightarrow & (d,d,e) \cr
f & \rightarrow & () \cr
\end{array}
]Please note that in an adjacency list representation of a digraph
* the order of the neighbors in each list is irrelevant, all that counts is the number of occurrences of each neighbor,
* if a vertex is in its own list, this indicates a loop,
* multiple occurrences of nodes in the list indicate multiple edges,
* empty lists
(())indicate isolated nodes (like for the node
(f)`). 
see also adjacency matrix. ↩