Graph theory and networks

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Graph theory and networks
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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).

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Special Graphs

• The complete graph Kn is a simple graph
  consisting of n vertices with each joined to each
  of the others by an edge. Each vertex in Kn has
  degree n 1.
• The complete bi-partite graph Km,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.

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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.

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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 its 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.

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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.

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Planar Graphs

• A graph is called planar if it can be drawn in
  the plane with no two edges crossing (except at a
  node).
• Eulers theorem
• For any connected graph drawn in the plane with R
  regions, N nodes and A arcs
• R N A 2