CSCI 4260 MATH 4150

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CSCI 4260MATH 4150

• GRAPH THEORY

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Last time, I forgot to mention.

• Email mylastname _at_ cs.rpi.edu
• Webpagehttp//www.cs.rpi.edu/isler
• Course webpagehttp//www.cs.rpi.edu/isler/teachi
  ng/spring06/

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Degree of a vertex

• Recall the definition of neighborhood from last
  lecture.
• Definitions
• degree of a vertex
• ?(G)
• ?(G)

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Theorem

• Let G (V,E) with E m. We have

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For bipartite G (U, W, E)
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Theorem

• Every graph has an even number of odd vertices
  (vertices with odd degree).

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Global properties from local investigation

• Connectivity of a graph. Local or global
  property?
• Suppose we are allowed to investigate only one
  vertex at a time.
• Can we say anything about the connectivity?

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Lets start with some sufficient conditions

• What is a
• Necessary condition
• Sufficient condition
• If G contains a vertex with degree n-1, then G is
  connected.
• How about n-2?

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Theorem

• If

• for every u,v ? V then
• G is connected
• diam(G) lt 2

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Corollary

• If

• then
• G is connected

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Theorem

• If

FALSE

• for every u,v ? V then
• G is connected

This means that our bound of n-1 is sharp (or
tight)
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Regular graph

• A graph is k-regular if every vertex has exactly
  k neighbors
• Remember our theoremno graph contains an odd
  number of odd vertices
• What does this say about regular graphs?
• Can you have a k regular graph for arbitrary n?

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k-regular graph

• We can not have k and n both odd.
• Can you think of other constraints?
• It turns out that this is the only constraint we
  have

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Theorem

• There exists a k-regular graph of order n if and
  only if at least one of r and k is even

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Theorem

• For every graph G and every integer r ??(G),
  there exists an r-regular graph containing G as
  an induced graph.

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r 3 G as given below
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Not so regular graphs

• Degree sequences
• graphical sequence
• How do we know if a given sequence is graphical?

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Theorem

• Lets d1, d2, , dn be non-increasing sequence.
• Constructs1 d2-1,d3-1, , dd11, d12,, dn
• s is graphical iff s1 is graphical

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If direction s1 is graphical

• Lets d1, d2, , dn be non-increasing sequence.
• Constructs1 d2-1,d3-1, , dd11, dd12,, dn
• s is graphical iff s1 is graphical

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Only if direction s is graphical

• Lets d1, d2, , dn be non-increasing sequence.
• Constructs1 d2-1,d3-1, , dd11, dd12,, dn
• s is graphical iff s1 is graphical

• Claim among all graphs with degree sequence s,
  there is one where the vertex with degree d1is
  adjacent to vertices with degrees d2, , dd11

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Proof of the claim. Suppose not
We have dr gt ds
vr
Hence there must be a third vertex, say vt, which
is adjacent to vr but not to vs
vs
vt
v1
Among all graphs with degree sequence of s, pick
the one such that the sum of the degrees of the
neighbors of v1 is max.
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Proof of the claim. Suppose not
Lets reorder the edges
vr
vs
vt
v1
What happened to the sum of the degrees of the
neighbors of v1?
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Irregular graphs

• Definition no two vertices have the same degree
• Can you come up with one?

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Theorem

• Irregular graphs dont exist.

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Quiz

• Please take out a sheet of paper and put your
  name.

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Theorem

• G is a connected graph of order gt 3
• if and only if
• it contains two vertices u,v such that G-u and
  G-v are connected.