Graph Terminology
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Def: A GRAPH,  G is a pair G=(V,E) where
      * V is a finite nonempty set (the "vertices", or "nodes")
      * E is a set of 2-element subsets of V (the "edges") 


Lots of definitions based on this
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* If {v,w} \in E then v and w are ADJACENT.
  Adjacency is a relation on V x V, and we may right v-w 
  instead of {v,w} \in E.
* If e={v,w} is an edge of E, then edge e is INCIDENT to v.
* The NEIGHBOR SET of v, N(v), is the set of vertices adjacent to v:
  N(v) = {w\in V: {v,w} in E}
* the DEGREE of v, d(v), is |N(v)|.
* a PATH in G is a sequence v_1, v_2, ..., v_k such that
  {v_i,v_{i+1}} in E for all 1 <= i <= k-1.
* Write v --- w if there is  path from v to w.  This is an equivalence
  relation.  
  Equivalently: v---w is the reflexive/transitive closure of the 
  adjacency relation, - .
* The blocks of v---w are the COMPONENTS of G.  
  In other words, v and w are in the same component if there is 
  a path from v to w.
* A CYCLE in G is a sequence of 3 or more vertices C=(v_1, ..., v_k)
  such that v_i - v_{i+1} for each 1<= i < j, and v_1 - v_k.
  Sometimes we think of a cycle as containing edges: the cycle C contains 
  the edges {v_1,v_2}, {v_2,v_3}, ..., {v_{k-1},v_k}, {v_1,v_k}.  
  The vertices of C is the SET of vertices that appear in 
  the sequence (v_1, ..., v_k).   The same terminology applies to the
  vertices and edges of a path P.
* A graph is ACYCLIC if it has no cycle.
* A TREE is a connected acyclic graph.
* A FOREST is a acyclic graph (ie, a bunch of trees).
* Let G=(V,E) be a tree.  Then a LEAF of G is a node of degree 1.
  An isolated node is also a leaf.
* A graph G=(V,E) on m edges is EULERIAN if it has a 
  cycle C=(v_1,..., v_m) where the edges of C are E (that is, each edge
  is used once and only once in C).
* A graph G=(V,E) on n vertices is HAMILTONIAN if it has a 
  cycle C=(v_1,..., v_n) where the vertices of C are exactly V (that is,   
  each vertex is used once and only once). 



Simple results
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Proposition: Every tree has a leaf.

Proposition: A tree on n vertices has n-1 edges.

Proposition: A connected graph on n vertices and n-1 edges is
             A tree  (that is, it is acyclic).

Proposition: A graph on n vertices has at most C(n,2) = n(n-1)/2 edges.

Proposition: Let G=(V,E) be a graph on n vertices and m edges.  
             Then m = (1/2) \sum_{v\in V} d(v)

Theorem [Euler]: Let G be a graph on 3 or more vertices.  
             Then G is Eulerian iff each vertex of G is of even degree.