1 Section 8.2 Graph Terminology

2 Terms related to undirected graphs Adjacent: 2 vertices u & v in an undirected graph G are adjacent (neighbors) in G if there is an edge {u,v} Incident, connect: if edge e = {u,v}, e is incident with vertices u & v, and e connects u and v Endpoints: vertices u & v are endpoints of edge e

3 Terms related to undirected graphs Degree of vertex in an undirected graph is the number of edges incident with it –loops count twice –degree of vertex v is denoted deg(v) Example: what are the degrees of the vertices in this graph?

4 Terms related to undirected graphs Isolated vertex (like e in previous example) has degree 0, not adjacent to any other vertex Pendant vertex (like d in previous example) - adjacent to exactly one vertex The sum of the degrees of all vertices in a graph is exactly twice the number of edges in the graph

5 Handshaking Theorem Let G=(V,E) be an undirected graph with e edges; then 2e = Note that sum of degrees of vertices is always even; this leads to the theorem: An undirected graph has an even number of vertices of odd degree

6 Terminology related to directed graphs Let G be a directed graph, with an edge e=(u,v): –u is initial vertex –v is terminal, or end vertex –u is adjacent TO v –v is adjacent FROM u –initial & terminal vertex of a loop are the same

7 Terminology related to digraphs In-degree of vertex v, denoted deg - (v), is the number of edges with v as the terminal vertex Out-degree of vertex v, denoted deg + (v), is the number of edges with v as the initial vertex A loop has one of each

8 Terminology related to digraphs The sum of in-degrees is equal to the sum of out- degrees Both are equal to the number of edges in the graph: –Let G=(V,E) be a directed graph; then The undirected graph that results from ignoring arrows in a directed graph is called the underlying undirected graph

9 Classes of simple graphs Complete graphs: complete graph on n vertices, denoted K n, is the simple graph that contains exactly one edge between each pair of distinct vertices; Examples:

10 Classes of Simple Graphs Cycle: C n, where n  3, consists of n vertices v 1, v 2, …, v n and edges {v 1, v 2 }, {v 2, v 3 }, …, {v n-1, v n } Examples:

11 Classes of Simple Graphs Wheel: a cycle with an additional vertex, which is adjacent to all other vertices; example:

12 Classes of Simple Graphs n-Cube: denoted Q n, is a graph representing the 2 n bit strings of length n: 2 vertices are adjacent if and only if the bit strings they represent differ in exactly one position. Examples:

13 Bipartite Graphs A simple graph G is called bipartite if its vertex set V can be partitioned into 2 disjoint non-empty sets V 1 and V 2 such that: –every edge connects a vertex in V 1 with a vertex in V 2 –no edge connects 2 vertices in V 1 or in V 2

14 Example: lining up 1st graders CateJohn MaryMark Mary AnnTerry Mary PatTim MarieJimmy

15 Example - cycle C6 is bipartite; can partition its vertex set into 2 distinct sets: –V 1 = {v 1, v 3, v 5 ) –V 2 = {v 2, v 4, v 6 } –with every edge connecting a vertex in V 1 with one in V 2

16 Examples Graph at left is not bipartite; to divide into 2 sets, one set must include 2 vertices and to be bipartite, those vertices must not be connected But every vertex in this graph is connected to 2 others This graph, on the other hand, is bipartite; the two sets are V 1 = {v 1, v 3, v 5 } and V 2 = {v 2, v 4, v 6 } Note that the definition doesn’t say a vertex in one set can’t connect to more than one vertex in the other - only that each in one must connect to one in the other, and no vertex in a set can connect to a vertex in the same set

17 Examples: are they bipartite?

18 Complete Bipartite Graphs Denoted K m,n is the graph that has its vertex set partitioned into 2 subsets of m vertices and n vertices There is an edge between 2 vertices if and only if one vertex is in the first subset and the other is in the second subset

19 Examples

20 Applications of Special Types of Graphs: LAN topology A star is a complete bipartite K 1,n graph: A ring is an n-cycle: A redundant network may have both a central hub and a ring, forming a wheel:

21 Subgraph Graph obtained by removing vertices and their associated edges from a larger graph; more formally: Subgraph of G=(V,E) is H=(W,F) where W  V and F  E

22 Subgraph Example

23 Can combine graphs, forming a union Let G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) The union, G 1  G 2 = (V 1  V 2, E 1  E 2 )

24 Section 8.2 Graph Terminology