Graph theory and networks

Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins one vertex to another, or a vertex to itself. A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins one vertex to another, or a vertex to itself.  The degree or order of a node is the number of ends of arcs at the vertex.  A connected graph is one where every vertex is linked (by a single arc or a sequence of arcs) to every other.  A subgraph of a graph is another graph that can be seen within it (i.e. another graph consisting of some of the original vertices and arcs).

Special Graphs  The complete graph K n is a simple graph consisting of n vertices with each joined to each of the others by an edge. Each vertex in K n has degree n – 1.  The complete bi-partite graph K m,n consists of two groups of vertices, illustrated with m vertices in one group and n in the other. Each of the m vertices is joined to each of the n vertices.  A connected graph with n vertices and n-1 arcs is called a tree.

Paths and cycles  A walk is a sequence of edges such that the end node of one edge in the sequence is the start node of the next edge in the sequence.  A trail is a walk such that no edge is included more than once (in either direction).  A path is a trail such that no vertex is visited more than once (except that the first vertex may be the same as the last).  A walk, trail or path is closed if the first vertex is the same as the last. A cycle (or circuit) is a closed path. If a cycle visits every node in the graph it is known as a Hamiltonian cycle.

Eulerian and semi-Eulerian graphs  A trail that uses all the edges of a graph is called a Eulerian trail.  If a graph possesses a closed Eulerian trail, then the graph itself is called Eulerian (i.e. a graph is called Eulerian if it is possible to start at a node, traverse each arc exactly once and end up where you started). This is possible if and only if every node has even order.  If a graph possesses an Eulerian trail that is not closed, the graph is called semi-Eulerian (i.e. a graph is semi-Eulerian if it’s possible to start at a node, traverse each arc exactly once and end up somewhere different to the starting point). This is possible if and only if the graph has exactly two odd nodes.

Result  In any graph, the sum of all the degrees = 2 × no. of edges. [By adding up all the degrees you are in effect counting all the ends of the edges. Since each edge has 2 ends, the total number of ends will be twice the number of edges]. [By adding up all the degrees you are in effect counting all the ends of the edges. Since each edge has 2 ends, the total number of ends will be twice the number of edges].

Planar Graphs  A graph is called planar if it can be drawn in the plane with no two edges crossing (except at a node).  Euler’s theorem For any connected graph drawn in the plane with R regions, N nodes and A arcs: R + N = A + 2