Subsections

• Adjacency, Degree and Distance Functions for a Graph
• Adjacency and Degree Functions for a Digraph

Given a graph G, and a non-negative integer n, return the set of all vertices of G that have degree equal to n.

Given a vertex u belonging to a graph G, and a non-negative integer n, return the set of vertices of G that are at distance n or less from u.

Given a bipartite graph G, return its two partite sets in the form of a pair of subsets of V(G).

Given a vertex u of the graph G, return the degree of u, ie the number of edges incident to u.

Given a graph G such that the maximum degree of any vertex of G is r, return a sequence D of length r + 1, such that D[i], 1 <= i <= r + 1, is the number of vertices in G having degree i - 1.

Given two vertices u and v belonging to a graph G, return the length of a shortest path from u to v. If u and v lie in different components, the value -1 is returned.

Given an edge e belonging to the graph G, return a set containing the two end vertices of e.

The set of all edges incident with the vertex u.

Maxdeg(G) : GrphUnd -> RngIntElt, GrphVert

The maximum of the degrees of the vertices of G. This function returns two values: The maximum degree, and a vertex of G having that degree.

Mindeg(G) : GrphUnd -> RngIntElt, GrphVert

The minimum of the degrees of the vertices of G. This function returns two values: The minimum degree, and a vertex of G having that degree.

Neighbors(u) : GrphVert -> { GrphVert }

Given a vertex u of the graph G, return the set of vertices of G that are adjacent to u.

Given a vertex u belonging to the graph G, and a non-negative integer n, return the set of vertices of G that are at distance n from u.

Given a regular graph G, return the valence of G (the degree of any vertex).

Given vertices u and v belonging to a digraph G, return the value true if there is an edge directed from vertex u to vertex v, otherwise false.

e adj f : GrphEdge, GrphEdge -> BoolElt

True if the edge e is adjacent to the edge f, where e and f are edges belonging to the digraph G, otherwise false.

Given vertices u and v belonging to a digraph G, return the value true if there is not an edge directed from vertex u to vertex v, otherwise false.

e notadj f : GrphEdge, GrphEdge -> BoolElt

True if the edge e is not adjacent to the edge f, where e and f are edges belonging to the digraph G, otherwise false.

True if the vertex u is an endpoint of the edge e, where u and e belong to the digraph G, otherwise false.

Given a digraph G, and a non-negative integer n, return the set of all vertices of G that have total degree equal to n.

Given a vertex u belonging to a connected digraph G, and a non-negative integer n, return the set of vertices of G that are at distance n or less from u. Thus, Ball(u, n) is the set of vertices of G that are connected to u by a directed path of length at most n.

Given a vertex u belonging to the digraph G, return the total degree of u, i.e. the sum of the in-degree and out-degree for u.

Given an edge e belonging to the digraph G, return a sequence of length two whose first component is the first vertex of e, and whose second component is the second vertex of e.

The number of edges directed into the vertex u belonging to the directed graph G.

InNeighbors(u) : GrphVert -> { GrphVert }

Given a vertex u of the digraph G, return the set containing all vertices v such that vu is an edge in the digraph, i.e. the starting points of all edges that are directed into the vertex u.

Maxdeg(G) : GrphDir -> RngIntElt, GrphVert

The maximum total degree of the vertices of the digraph G. This function returns two values: The maximum total degree, and the first vertex of G having that degree.

Maxindeg(G) : GrphDir -> RngIntElt, GrphVert

The maximum indegree of the vertices of the digraph G. This function returns two values: The maximum indegree, and the first vertex of G having that degree.

Maxoutdeg(G) : GrphDir -> RngIntElt, GrphVert

The maximum outdegree of the vertices of the digraph G. This function returns two values: The maximum outdegree, and the first vertex of G having that degree.

Mindeg(G) : GrphDir -> RngIntElt, GrphVert

The minimum total degree of the vertices of the digraph G. This function returns two values: The minimum total degree, and the first vertex of G having that degree.

Minindeg(G) : GrphDir -> RngIntElt, GrphVert

The minimum indegree of the vertices of the digraph G. This function returns two values: The minimum indegree, and the first vertex of G having that degree.

Minoutdeg(G) : GrphDir -> RngIntElt, GrphVert

The minimum outdegree of the vertices of the digraph G. This function returns two values: The minimum outdegree, and the first vertex of G having that degree.

The number of edges of the form uv where u is a vertex belonging to the directed graph G.

OutNeighbors(u) : GrphVert -> { GrphVert }

Given a vertex u of the digraph G, return the set of vertices v of G such that uv is an edge in the graph G, i.e. the set of vertices v that are the end vertices of edges directed from u to v.