Graphs isomorphism, paths, cycles, connectivity

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Graphs isomorphism, paths, cycles, connectivity
Homework 9.3 34, 36, 38, 40, 42 9.4 2,
4, 6, 18, 20, 22
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Isomorphism

• Graphs are isomorphic if they have the same
  structure
• 1-to-1 mapping of vertices
• 1-to-1 mapping of edges

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Isomorphism

• Isomorphic graphs will have the same adjacency
  matrix under some reordering of the vertices
• abcdef ? bacdfe

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Path

• A path begins at vertex v0, follows an edge e1 to
  v1, follows another edge to v2
• A path is represented without edges (especially
  when there are no parallel edges)
• (v0, v1, v2, vn)
• Said to be of length n
• A path on a vertex itself is of length 0
• A simple path from v1 to vx is a path with no
  repeated edges.

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Circuit Definitions

• A circuit/cycle is a path of nonzero length from
  v to v.
• A simple circuit is a circuit from v to v with no
  repeated edges

path (b,c,d,e,a,b) is a circuit path
(a,b,a,e,d,c,a) is a circuitpath (b,a,c,d,e,a,b)
is a circuit path (a,d,c,a,e,d,a) is a
circuit path (c,d,e,a,b,c) is a circuit
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Connected Graph

• A graph G is connected if given any vertices v1
  and v2 in G, there is a path from v1 to v2.

Connected
b
c
Not Connected
e
f
b
c
f
a
d
a
d
connected components
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Counting paths -- adjacency matrix

• Given M an adjacency matrix and n gt0
• Mn will give the number of paths of length n
  between vertices
• For a simple graph, the entries on the main
  diagonal of M2 give the degrees of the vertices

B
A
C
D
E

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Next week Euler Circuit

• A Euler Circuit in a graph G is a simple circuit
  that includes all edges and vertices of G.
• A Euler Path in a graph G is a simple path that
  includes all edges and vertices of G.

Visit each edge once
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Next week Hamiltonian circuit

• a circuit where each vertex in G is used
  exactly once

Visit each vertex once
Hamiltonian Euler circuit
Hamiltonian circuit