On the Edge-Balance Index of Flux Capacitor and L-Product of Star by Cycle Graphs

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On the Edge-Balance Index of Flux Capacitor and
L-Product of Star by Cycle Graphs

• Meghan Galiardi, Daniel Perry, Hsin-hao Su
• Stone hill College

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Labeling of the Graphs

• The edges of the graph are labeled by the group
  Z20, 1
• The vertices are labeled according to the
  adjacent edges
• Vertex labeled 0 if the number of edges adjacent
  labeled 0 is greater than the edges labeled 1
• Vertex labeled 1 if the number of edges adjacent
  labeled 1 is greater than the edges labeled 0
• Vertex unlabeled if the number of edges adjacent
  labeled 0 is equal to the edges labeled 1

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Edge-Friendly Graphs

• The graphs are said to be edge-friendly if the
  number of 1-edges and 0-edges differ by no more
  than 1.
• e(0)-e(1) 1

Example total edges 12 e(0) 6 e(1) 6
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Edge-Balance Index Set

• The edge-balance index is the difference between
  0-vertices and 1-vertices
• EBI v(0) v(1)
• The edge-balance index set for graph G is the set
  of all possible edge-balance indices that G can
  have
• We looked for the edge-balance index sets of two
  types of graphs

Example v(0) 3 v(1) 2 EBI 1
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Flux Capacitor Graphs

• Definition A flux capacitor graph is composed
  of two different types of graphs, a star graph
  and a cycle graph. A star graph, St(n), consists
  of a center vertex and n surrounding vertices
  each connected to the center. A cycle graph, Cm,
  consists of m vertices each connected to 2 others
  to form a cycle where m3. A flux capacitor
  graph, FC(n, m), is a St(n) graph where on each
  outer vertex there is a graph Cm.

St(3)
C3
FC(3, 3)
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Theorems

• EBI(FC(n, m))
• 0, 1, , n-1 if m is odd
• 0, 1, , n if n is odd and m is even
• 0, 1, , n-1 if n is even and m is even

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How we proved it

• Started with FC(n, 3) and FC(n, 4)
• First we looked for the most efficient way to
  label the graphs as to achieve the highest EBI
• From the highest EBI we looked at how we can
  rearrange the graphs to decrement the EBI by 1
• We rearranged the graphs as many times as it took
  to achieve EBI from the highest all the way to 0
• The results we found also generalized for any
  FC(n, m)

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FC(n, 3)EBI(FC(n, 3)) 0, 1, , n-1
To decrease the EBI by one, simply switch a
0-edge and a 1-edge on one of the cycles. This
changes the 0-vertex to a 1-vertex and adds an
additional 0-vertex. Since v(0) was greater that
v(1). This change causes the EBI to decrease by 1.
Most efficient way to label is to label the star
with all 1-edges and then alternate the cycle
with 0 and 1-edges. This creates EBI n-1.
v(0) v(1) 1
v(0) v(1) 0
EBI 0, 1
Note We assumed that e(0)e(1) and by our
labeling v(0)v(1). The opposite can also be
assumed, but the results for the EBI will still
be the same so we only have to look at one case
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FC(n, 4) if n is even EBI(FC(n, 4)) 0, 1, ,
n-1
When n is even the most efficient way to label
the graph is shown below. This creates EBI n-1.
Again the edges can be rearranged to decrement
the EBI by 1 each time, and all the EBI from n-1
all the way to 0.
v(0) v(1) 1
v(0) v(1) 0
EBI 0, 1
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FC(n, 4) if n is odd EBI(FC(n, 4)) 0, 1, ,
n
The same can be done when n is odd, there is just
a slightly different way of labeling the graph
for the highest EBI. This creates EBI n-1.
Again the edges can be rearranged to decrement
the EBI by 1 each time, and all the EBI from n-1
all the way to 0.
v(0) v(1) 3
v(0) v(1) 2
v(0) v(1) 1
v(0) v(1) 0
EBI 0, 1, 2, 3
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FC(n, m)
The results for EBI(FC(n, m)) generalize from
FC(n, 3) and FC(n, 4) Example EBI(FC(4, 7))
0, 1, 2, 3
v(0) v(1) 3
v(0) v(1) 2
v(0) v(1) 1
v(0) v(1) 0
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L-Product of Cycle by Star

• Definition An L-product of cycle by star graph
  is the same as a flux capacitor graph, the only
  difference being there is an additional cycle,
  Cm, on the center vertex of the star. It is
  represented as St(n)xLCm.

St(3)xLC3
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Theorems

• EBI(St(n)xLCm)
• 0, 1, , n1 if m is odd
• 0, 1, , n1 if n is odd and m is even
• 0, 1, , n if n is even and m is even

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How we proved it

• We started with FC(n1, m). By removing 1 edge
  and merging 2 vertices we can create St(n)xLCm
• When m is odd, EBI(FC(n, m)) 0, 1, , n-1
• EBI(FC(n1, m)) 0, 1, , n
• FC(n1, m) has an even number of edges. Removing
  an edge will keep the graph edge friendly and
  cause the EBI to change at most by 1
• So EBI(St(n)xLCm) 0, 1, , n1 when m is odd

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How we proved it

• When n1 is even and m is even,
• When n is even EBI(FC(n, m)) 0, 1, , n-1
• When n1 is even EBI(FC(n1, m)) 0, 1, , n
• FC(n1, m) has an even number of edges. Removing
  an edge will keep the graph edge friendly and
  cause the EBI to change at most by 1
• Starting with n1 even and removing the edge
  makes n odd, while m stays even
• So EBI(St(n)xLCm) 0, 1, , n1 when n is odd
  and m is even

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How we proved it

• When n1 is odd and m is even,
• FC(n1, m) has an odd number of edges. Removing
  an edge may not keep the graph edge friendly so
  the previous method does not work
• Results from the flux capacitor graphs could not
  be used so we created a most efficient way to
  label
• It was found EBI(St(n)xLCm) 0, 1, , n when
  n is even and m is even

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St(2)xLC3
v(0) v(1) 3
v(0) v(1) 2
v(0) v(1) 1
v(0) v(1) 0
n is even, m is odd EBI(St(n)xLCm) 0, 1, ,
n1 EBI 0, 1, 2, 3
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Conclusions

• EBI(FC(n, m))
• 0, 1, , n-1 if m is odd
• 0, 1, , n if n is odd and m is even
• 0, 1, , n-1 if n is even and m is even
• EBI(St(n)xLCm)
• 0, 1, , n1 if m is odd
• 0, 1, , n1 if n is odd and m is even
• 0, 1, , n if n is even and m is even