Adjacency and Degree

The adjacency and degree functionalities that apply to simple graphs (see Adjacency and Degree) similarly apply to multigraphs.

Adjacency and Degree Functions for Multigraphs

Degree(u): GrphVert -> RngIntElt

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

Alldeg(G, n): GrphMultUnd, RngIntElt -> { GrphVert}

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

MaximumDegree(G): GrphMultUnd -> RngIntElt, GrphVert

Maxdeg(G): GrphMultUnd -> RngIntElt, GrphVert

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

MinimumDegree(G): GrphMultUnd -> RngIntElt, GrphVert

Mindeg(G): GrphMultUnd -> RngIntElt, GrphVert

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

DegreeSequence(G): GrphMultUnd -> [ { GrphVert} ]

Given a multigraph \(G\) such that the maximum degree of any vertex of \(G\) is \(r\), return a sequence \(D\) of length \(r+1\), such that \(D[i]\), \(1 \leq i \leq r+1\), is the number of vertices in \(G\) having degree \(i-1\).

Neighbours(u): GrphVert -> { GrphVert}

Neighbors(u): GrphVert -> { GrphVert}

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

IncidentEdges(u): GrphVert -> { GrphEdge}

Given a vertex \(u\) of a graph \(G\), return the set of all edges incident with the vertex \(u\).

Adjacency and Degree Functions for Multidigraphs

InDegree(u): GrphVert -> RngIntElt

The number of edges directed into the vertex \(u\) belonging to a multidigraph.

OutDegree(u): GrphVert -> RngIntElt

The number of edges of the form \([u, v]\) where \(u\) is a vertex belonging to a multidigraph.

MaximumInDegree(G): GrphMultDir -> RngIntElt, GrphVert

Maxindeg(G): GrphMultDir -> RngIntElt, GrphVert

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

MinimumInDegree(G): GrphMultDir -> RngIntElt, GrphVert

Minindeg(G)): GrphMultDir -> RngIntElt, GrphVert

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

MaximumOutDegree(G): GrphMultDir -> RngIntElt, GrphVert

Maxoutdeg(G): GrphMultDir -> RngIntElt, GrphVert

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

MinimumOutDegree(G): GrphMultDir -> RngIntElt, GrphVert

Minoutdeg(G): GrphMultDir -> RngIntElt, GrphVert

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

Degree(u): GrphVert -> RngIntElt

Given a vertex \(u\) belonging to the multidigraph \(G\), return the total degree of \(u\), i.e. the sum of the in–degree and out–degree for \(u\).

MaximumDegree(G): GrphMultDir -> RngIntElt, GrphVert

Maxdeg(G): GrphMultDir -> RngIntElt, GrphVert

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

MinimumDegree(G): GrphMultDir -> RngIntElt, GrphVert

Mindeg(G): GrphMultDir -> RngIntElt, GrphVert

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

Alldeg(G, n): GrphMultDir, RngIntElt -> { GrphVert}

Given a multidigraph \(G\), and a non–negative integer \(n\), return the set of all vertices of \(G\) that have total degree equal to \(n\).

DegreeSequence(G): GrphMultDir -> [ { GrphVert } ]

Given a multidigraph \(G\) such that the maximum degree of any vertex of \(G\) is \(r\), return a sequence \(D\) of length \(r + 1\), such that \(D[i]\), \(1 \leq i \leq r + 1\), is the number of vertices in \(G\) having degree \(i - 1\).

InNeighbours(u): GrphVert -> { GrphVert}

InNeighbors(u): GrphVert -> { GrphVert}

Given a vertex \(u\) of a multidigraph \(G\), return the set containing all vertices \(v\) such that \([v, u]\) is an edge in \(G\), i.e. the initial vertex of all edges that are directed into the vertex \(u\).

OutNeighbours(u): GrphVert -> { GrphVert}

OutNeighbors(u): GrphVert -> { GrphVert}

Given a vertex \(u\) of the multidigraph \(G\), return the set of vertices \(v\) of \(G\) such that \([u,v]\) is an edge in \(G\), i.e. the set of vertices \(v\) that are terminal vertices of edges directed from \(u\) to \(v\).

IncidentEdges(u): GrphVert -> { GrphEdge}

Given a vertex \(u\) of a graph \(G\), return the set of all edges incident with the vertex \(u\), that is, the set of all edges incident into \(u\) and incident from \(u\).