Graphs

Definition: A graph \(G = (V,E)\) consists of a nonempty set \(V\) of vertices (or nodes) and a set \(E\) of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.

Some Terminology

In a simple graph each edge connects two different vertices and no two edges connect the same pair of vertices.
A multigraph may have multiple edges connecting the same two vertices. When \(m\) different edges connect the vertices \(u\) and \(v\), we say that \(\{u, v\}\) is an edge with multiplicity \(m\).
An edge that connects a vertex to itself is called a loop.
A pseudograph may include loops, as well as multiple edges connecting the same pair of vertices.

Definition: An directed graph (or digraph) \(G = (V,E)\) consists of a nonempty set of \(V\) vertices (or nodes) and a set of directed edges (or arcs). Each edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair \((u,v)\) is said to start at u and end at v.

A simple directed graph has no loops and no multiple edges
A directed multigraph may have multiple directed edges. When there are \(m\) directed edges from the vertex \(u\) to the vertex \(v\), we say that \((u,v)\) is an edge of multiplicity \(m\).

Type	Edges	Multiple Edges	Loops
Simple graph	Undirected	No	No
Multigraph	Undirected	Yes	No
Pseudograph	Undirected	Yes	Yes
Simple directed graph	Directed	No	No
Directed multigraph	Directed	Yes	Yes
Mixed graph	Both	Yes	Yes

Definition: An adjacency list can be used to represent a graph with no multiple edges by specifying the vertices that are adjacent to each vertex in the graph

Definition: Suppose that \(G = (V,E)\) is a simple graph where \(\lvert V \rvert = n\). Arbitrarily list the vertices of \(G\) as \(v_1, v_2, \hdots, v_n\). The adjacency matrix \(A_G\) of \(G\), with respect to the listing of vertices, is the \(n \times n\) zero-one matrix with 1 as its \((i,j)th\) entry when \(v_i\) and \(v_j\) are adjacent, and 0 as its \((i,j)th\) entry when they are not adjacent.

Adjacency matrices can also be used to represent graphs with loops and multiple edges.
A loop at the vertex \(v_i\) is represented by a 1 at the \((i,j)th\) position of the matrix.
When multiple edges connect the same pair of vertices \(v_i\) and \(v_j\), the \((i,j)th\) entry equals the number of edges connecting the pair of vertices.

Adjacency matrices can also be used to represent directed graphs. The matrix for a directed graph \(G = (V,E)\) has a 1 in its \((i,j)th\) position if there is an edge from \(v_i\) to \(v_j\) where \(v_1, v_2, \hdots, v_n\) is a list of vertices.
The adjacency matrix for a directed graph does not have to be symmetric because there may not be an edge from \(v_i\) to \(v_j\) when there is an edge from \(v_j\) to \(v_i\).
To represent directed multigraphs, the value of \(a_{ij}\) is the number of edges connecting \(v_i\) to \(v_j\).

Informal Definition: A path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph. As the path travels along its edges, it visits the vertices along this path, that is, the endpoints of these.

Definition: Let \(n\) be a nonnegative integer and \(G\) an undirected graph. A path of length \(n\) from \(u\) to \(v\) in \(G\) is a sequence of \(n\) edges \(e_1, \ldots, e_n\) of \(G\) for which there exists a sequence \(x_0 = u, x_1, \ldots, x_{n-1}, x_n = v\) of vertices such that \(e_i\) has, for \(i = 1, \ldots, n\), the endpoints \(x_{i-1}\) and \(x_i\).
- When the graph is simple, we denote this path by its vertex sequence \(x_0, x_1, \ldots, x_n\) (since listing the vertices uniquely determines the path).
- The path is a circuit if it begins and ends at the same vertex (\(u = v\)) and has length greater than zero.
- The path or circuit is said to pass through the vertices \(x_1, x_2, \ldots, x_{n-1}\) and traverses the edges \(e_1, \ldots, e_n\).
- A path or circuit is simple if it does not contain the same edge more than once.

Definition: An undirected graph is called connected if there is a path between every pair of vertices. An undirected graph is that is not connected is called disconnected. We say that we disconnect a graph when we remove vertices or edges, or both, to produce a disconnected subgraph.
Definition: A connected component of a graph \(G\) is a connected subgraph of \(G\) that is not a proper subgraph of another connected subgraph of \(G\).

Definition: An directed graph is called strongly connected if there is a path from \(a\) to \(b\) and a path from \(b\) to \(a\) whenever \(a\) and \(b\) are vertices in the graph.
Definition: A directed graph is weakly connected if there is a path between every two vertices in the underlying undirected graph, which is the undirected graph obtained by ignoring the directions of the edges of the directed graph.

Definition: A tree is a connected undirected graph with no simple circuits.
Definition: A forest is a graph that has no simple circuit but is not connected. Each of the connected components in a forest is a tree.

A rooted tree is a tree in which one vertex has been designated as the root and every edge is directed away from the root.
An unrooted tree is converted into different rooted trees when different vertices are chosen as the root.

If \(v\) is a vertex of a rooted tree other than the root, the parent of \(v\) is the unique vertex \(u\) such that there is a directed edge from \(u\) to \(v\). When \(u\) is a parent of \(v\), \(v\) is called the child of \(u\). Vertices with the same parent are called siblings.
The ancestors of a vertex are the vertices in the path from the root to this vertex, excluding the vertex itself and including the root.
The descendants of a vertex \(v\) are those vertices that have \(v\) as an ancestor.
A vertex of a rooted tree with no children is called a leaf. Vertices that have children are called internal vertices.
If \(a\) is a vertex in a tree, the subtree with \(a\) as its root is a tree consisting of \(a\) and its descendants and all edges incident to these descendants.

Definition: Let \(G\) be a simple graph. A spanning tree of \(G\) is a subgraph of \(G\) that is a tree containing every vertex of \(G\).

Definition: A weighted graph is a graph in which the edges have been assigned weights where the weights are numerical values.
The shortest path problem is the problem of finding a path between two vertices in a weighted graph such that the sum of the weights of its constituent edges is minimized.
Dijkstra’s algorithm solves the problem of finding the shortest path from a source vertex \(v \in V\) to all other vertices in the graph assuming that the weights are nonnegative.