MA375 Ch. 10: Graphs and Graph Models

directed edge

associated with ordered pair (u, v) where edge starts at u and ends at v

isolated vertex

degree of vertex is 0

initial vertex

for edge (u, v), u is the initial vertex

k-connected

if k(G) >= K

union of two simple graphs

G1 U G2 = (V1 U V2, E1 U E2)

subgraph of G = (V, E)

H = (W, F), where W is a subset of V and F of E; PROPER subgraph of G if H != G

The Handshaking Theorem

If G = (V , E) be an undirected graph with m edges, then 2m = v∈V deg(v); the sum of all the degrees in G is twice the number of edges

G = (V, E)

V = vertices, E = edges, edge connects endpoints

bipartite graph

V can be divided into two disjoint sets such that every edge in the graph connects a vertex in V1 to a vertex in V2 (no edge connects V1 to V1, same for V2)

directed graph (digraph)

a graph in which each edge is directed from one vertex to another (or the same) vertex; may contain loops and multiple edges

undirected graph

a graph in which the edges have no direction

sparse graph

a graph that contains relatively few edges; each vertex has a degree not exceeding c, where c is a constant far less than n; easier to use adjacency list to represent

complete bipartite graph (Km,n)

a graph that has its vertex set partitioned into two sets of m and n vertices, with an edge between each pair of vertices in the opposite sets

pseudograph

a graph that includes loops and/or multiple edges connecting the same pair of vertices or a vertex to itself

dense graph

a graph with a lot of edges; easier to use adjacency matrix to represent

mixed graph

a graph with directed and undirected edges; multiple edges and loops allowed

directed multigraph

a graph with directed edges that may contain multiple directed edges; each edge is associated with ordered pair (u, v) and multiplicity m

multigraph

a graph with multiple edges connecting the same vertices (term: multiplicity m)

adjacency list

a list that specifies vertices that are adjacent to each vertex of a simple graph (no multiple edges)

nonseparable graph

a nontrivial connected graph with no cut-vertices

graph invariant

a property preserved by isomorphism of graphs; used to prove nonisomorphism; example: isomorphic simple graphs must have the same number of edges and the same number of vertices

noncomplete graph

a simple graph for which there is at least one pair of distinct vertices not connected by an edge

matched

a vertex that is the endpoint of an edge of a matching M; otherwise unmatched

cut vertex

a vertex whose deletion produces a subgraph that is not connected

wheel (Wn)

add an additional vertex to Cn and have an edge connecting it to every vertex

adjacency matrix A of G (AG)

aij = 1 when {vi, vj} is an edge of G, 0 otherwise

loop

an edge that connects a vertex with itself

cut edge

an edge whose deletion increases the number of components

How many vertices of odd degree does an undirected graph have?

an even number

incidence matrix

an mxn matrix if |V| = n and |E| = m where the columns are represented by edges and rows by vertices; mij = 1 if edge ej is incident with vi; can represent multiple edges and loops

connectedness in undirected graphs

an undirected graph is connected if there is a path between every pair of distinct vertices

Hall's Marriage Theorem

bipartite graph G = (V, E) with bipartition (V1, V2) has complete matching from V1 to V2 if and only if |N(A)| >= |A| for all subsets A of V1; for any subset of V1, there must be more or the same amount of adjacent vertices as there are vertices.

simple graph

each edge connects two different vertices and no two edges connect the same pair of vertices

incident edge

edge e connects two vertices u and v in G; undirected graph

finite graphs

finite vertices, graphs

terminal/end vertex

for edge (u, v), v is the terminal or end vertex

counting paths between vertices

if A is the adjacency matrix for G, the number of different paths of length r from vi to vj is the i-jth entry of A^r

complete matching from V1 to V2

if every vertex in V1 is the endpoint of an edge in the matching; if |M| = |V1|

connected component

in an undirected graph, a connected component is a maximal set of vertices such that there is a path between every pair of vertices

deg-(v)

in-degree of a vertex v; the number of edges with v as terminal vertex

infinite graphs

infinite vertices and/or edges

a simple graph is partite if...

it is possible to assign two colors to vertices such that no adjacent vertices are the same color

edge and vertex connectivity

k(g) <= gamma(G) <= min deg(v)

maximum matching

matching with the largest number of edges

edge connectivity gamma(G)

minimum number of edges in an edge cut

cycle (Cn)

n >= 3; has n vertices; only edges between ordered vertices (n-1, n, n+1)

deg+(v)

out-degree of v; the number of edges with v as initial vertex

bipartition

pair (V1, V2) of graph V

simple path

path that does not contain an edge more than once

edge contraction

remove an edge e and its vertices u and v; replace u and v with a single new vertex w that connects to all other vertices that u and v did; result is NOT a subgraph

path

sequence of edges that begins at a vertex and travels to others along edges of the graph; denoted by x0, x1, ... , xn

complete graph on n vertices (Kn)

simple graph that contains exactly one edge for every pair of distinct vertices

How many edges does a graph with directed edges have?

sum(deg-(v)) = sum(deg+(v)) = E

deg(v)

the degree of a vertex in an undirected graph is the number of edges incident with it; a loop contributes twice to the degree

isomorphism

the function f from V1 to V2 with the property that a and b are adjacent in G1 and f(a) and f(b) are adjacent in G2, for all a and b in V1

vertex connectivity (k(G))

the minimum number of vertices in a vertex cut

N(v) (neighborhood of v)

the set of all neighbors of a vertex v of G(V, E); if A is a subset of V, N(A) is the set of all vertices adjacent to at least one vertex in A

vertex cut

the set of the cut vertices

matching M

the subset of the set E of edges of the graph such that no two edges are incident with the same vertex in a simple graph G = (V, E)

underlying undirected graph

the undirected graph that comes from ignoring directions on a directed graph; has same number of edges

isomorphic graphs

there is a one-to-one correspondence between two graph vertex sets that preserve edges

weakly connected directed graph

there is a path between every two vertices in the underlying undirected graph

adjacent/neighbor vertices

u and v are endpoints of an edge e in G; undirected graph

pendant vertex

vertex of degree 1

n-cube/n-dimensional hypercube (Qn)

vertices represent 2^n bit strings of length n

adjacent vertices for directed graphs

when (u, v) is an edge of a directed graph, u is said to be adjacent TO v and v is said to be adjacent FROM u

subgraph induced by a subset W of the vertex set V

when G is a simple graph, the edge set F contains an edge in E if and only if both endpoints of this edge are in W

strongly connected directed graph

when there's a path from a to be and from b to a whenever a and b are vertices in the graph