## Directed and Undirected Graphs

### What Is a Graph?

A graph is a collection of nodes and edges that represents relationships:

• Nodes are vertices that correspond to objects.

• Edges are the connections between objects.

• The graph edges sometimes have Weights, which indicate the strength (or some other attribute) of each connection between the nodes.

These definitions are general, as the exact meaning of the nodes and edges in a graph depends on the specific application. For instance, you can model the friendships in a social network using a graph. The graph nodes are people, and the edges represent friendships. The natural correspondence of graphs to physical objects and situations means that you can use graphs to model a wide variety of systems. For example:

• Web page linking — The graph nodes are web pages, and the edges represent hyperlinks between pages.

• Airports — The graph nodes are airports, and the edges represent flights between airports.

In MATLAB®, the graph and digraph functions construct objects that represent undirected and directed graphs.

• Undirected graphs have edges that do not have a direction. The edges indicate a two-way relationship, in that each edge can be traversed in both directions. This figure shows a simple undirected graph with three nodes and three edges.

• Directed graphs have edges with direction. The edges indicate a one-way relationship, in that each edge can only be traversed in a single direction. This figure shows a simple directed graph with three nodes and two edges.

The exact position, length, or orientation of the edges in a graph illustration typically do not have meaning. In other words, the same graph can be visualized in several different ways by rearranging the nodes and/or distorting the edges, as long as the underlying structure does not change.

#### Self-loops and Multigraphs

Graphs created using graph and digraph can have one or more self-loops, which are edges connecting a node to itself. Additionally, graphs can have multiple edges with the same source and target nodes, and the graph is then known as a multigraph. A multigraph may or may not contain self-loops.

For the purposes of graph algorithm functions in MATLAB, a graph containing a node with a single self-loop is not a multigraph. However, if the graph contains a node with multiple self-loops, it is a multigraph.

For example, the following figure shows an undirected multigraph with self-loops. Node A has three self-loops, while node C has one. The graph contains these three conditions, any one of which makes it a multigraph.

• Node A has three self-loops.

• Nodes A and B have five edges between them.

• Nodes A and C have two edges between them.

To determine whether a given graph is a multigraph, use the ismultigraph function.

### Creating Graphs

The primary ways to create a graph include using an adjacency matrix or an edge list.

One way to represent the information in a graph is with a square adjacency matrix. The nonzero entries in an adjacency matrix indicate an edge between two nodes, and the value of the entry indicates the weight of the edge. The diagonal elements of an adjacency matrix are typically zero, but a nonzero diagonal element indicates a self-loop, or a node that is connected to itself by an edge.

• When you use graph to create an undirected graph, the adjacency matrix must be symmetric. In practice, the matrices are frequently triangular to avoid repetition. To construct an undirected graph using only the upper or lower triangle of the adjacency matrix, use graph(A,'upper') or graph(A,'lower') .

• When you use digraph to create a directed graph, the adjacency matrix does not need to be symmetric.

• For large graphs, the adjacency matrix contains many zeros and is typically a sparse matrix.

• You cannot create a multigraph from an adjacency matrix.

For example, consider this undirected graph.

You can represent the graph with this adjacency matrix:

$\left(\begin{array}{ccc}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$

To construct the graph in MATLAB, input:

A = [0 1 2; 1 0 3; 2 3 0];
node_names = {'A','B','C'};
G = graph(A,node_names)
G =

graph with properties:

Edges: [3×2 table]
Nodes: [3×1 table]

You can use the graph or digraph functions to create a graph using an adjacency matrix, or you can use the adjacency function to find the weighted or unweighted sparse adjacency matrix of a preexisting graph.

#### Edge List

Another way to represent the information in a graph is by listing all of the edges.

For example, consider the same undirected graph.

Now represent the graph by the edge list

From the edge list it is easy to conclude that the graph has three unique nodes, A, B, and C, which are connected by the three listed edges. If the graph had disconnected nodes, they would not be found in the edge list, and would have to be specified separately.

In MATLAB, the list of edges is separated by column into source nodes and target nodes. For directed graphs the edge direction (from source to target) is important, but for undirected graphs the source and target node are interchangeable. One way to construct this graph using the edge list is to use separate inputs for the source nodes, target nodes, and edge weights:

source_nodes = {'A','A','B'};
target_nodes = {'B','C','C'};
edge_weights = [1 2 3];
G = graph(source_nodes, target_nodes, edge_weights);

Both graph and digraph permit construction of a simple graph or multigraph from an edge list. After constructing a graph, G, you can look at the edges (and their properties) with the command G.Edges. The order of the edges in G.Edges is sorted by source node (first column) and secondarily by target node (second column). For undirected graphs, the node with the smaller index is listed as the source node, and the node with the larger index is listed as the target node.

Since the underlying implementation of graph and digraph depends on sparse matrices, many of the same indexing costs apply. Using one of the previous methods to construct a graph all at once from the triplet pairs (source,target,weight) is quicker than creating an empty graph and iteratively adding more nodes and edges. For best performance, minimize the number of calls to graph, digraph, addedge, addnode, rmedge, and rmnode.

### Graph Node IDs

By default, all of the nodes in a graph created using graph or digraph are numbered. Therefore, you always can refer to them by their numeric node index.

If the graph has node names (that is, G.Nodes contains a variable Name), then you also can refer to the nodes in a graph using their names. Thus, named nodes in a graph can be referred to by either their node indices or node names. For example, node 1 can be called, 'A'.

The term node ID encompasses both aspects of node identification. The node ID refers to both the node index and the node name.

For convenience, MATLAB remembers which type of node ID you use when you call most graph functions. So if you refer to the nodes in a graph by their node indices, most graph functions return a numeric answer that also refers to the nodes by their indices.

A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0];
G = graph(A,{'a','b','c','d'});
p = shortestpath(G,1,4)
p =

1     3     4

However, if you refer to the nodes by their names, then most graph functions return an answer that also refers to the nodes by their names (contained in a cell array of character vectors or string array).

p1 = shortestpath(G,'a','d')
p1 =

1×3 cell array

{'a'}    {'c'}    {'d'}

Use findnode to find the numeric node ID for a given node name. Conversely, for a given numeric node ID, index into G.Nodes.Name to determine the corresponding node name.

### Modify or Query Existing Graph

After you construct a graph or digraph object, you can use a variety of functions to modify the graph structure or to determine how many nodes or edges the graph has. This table lists some available functions for modifying or querying graph and digraph objects.

 addedge Add one or more edges to a graph rmedge Remove one or more edges from a graph addnode Add one or more nodes to a graph rmnode Remove one or more nodes from a graph findnode Locate a specific node in a graph findedge Locate a specific edge in a graph numnodes Find the number of nodes in a graph numedges Find the number of edges in a graph edgecount Number of edges between specified nodes flipedge Reverse the direction of directed graph edges reordernodes Permute the order of the nodes in a graph subgraph Extract subgraph

See Modify Nodes and Edges of Existing Graph for some common graph modification examples.