Balance Index Set of Generalized Ear Expansion Hsin-hao Su Patrick Clark Dan Bouchard*
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Transcript of Balance Index Set of Generalized Ear Expansion Hsin-hao Su Patrick Clark Dan Bouchard*
Balance Index Set of Generalized Ear Expansion
Hsin-hao Su
Patrick Clark
Dan Bouchard*
Labeling the graphLet G be a simple graph with vertex set V(G) and edge set E(G)and let Z2 = {0,1}.
: 0 vertex
: 1 vertexI
A labeling f: V(G) Z2
which induces an edge partial labeling
f* : E(G) Z2 defined by
f*(uv) = f(u) iff f(u) = f(v), where u, v ∈ V(G).
0
1
f is called a friendly labeling if |vf (0) - vf (1)| ≤ 1
The BI(G), the balance index of G, is defined as: {|ef (0) - ef (1)| : the vertex labeling f is friendly.}
BI(G) = |1 – 1| = 0
Finding a Balance Index (BI): 0 vertex
: 1 vertexI • Create a friendly labeling Difference between 0 and 1 vertices less than or equal to 1
Five 0-verticesSix 1-vertices
Eleven total vertices
|5-6| = 1 ≤ 1
• Induce edge labeling Case 1: Two 0-vertices
Result: 0 edge Case 2: Two 1-vertices
Result: 1 edge Case 3: One of each
Result: Unlabeled edge
0 0
1
1
1
1• Balance Index is absolute value of difference between 0 and 1-edges
Two 0-edgesFour 1-edges
BI =|2-4| = 2
Generalized Ear Expansion
1 2
3
4
5
k2 = 2k1 = 3
k3 = 2
k5 = 2
ki = Number of ear
expansions on the corresponding edge i
Algebraic Equalities Adapted from
Kwong and Shiu
Even number of vertices
Number of 1-vertices in inner cycle = q
Using the corollary
Inner (blue) edges degree = n – 2q
Outer edges degree = 2q – n
Therefore, BI set is determined by labeling of inner vertices and red edges
Key results
• The balance index can be directly related to the degrees of the vertices
• Only the quantity of red edges connected to inner vertices are significant
• However, the labelingof the inner cycle’s vertices is also important
A closer look at a singleedge of the inner cycle
v1
v2
12
k2 = 2
Possibility 1: v1 and v2 are both 0-vertices
e(0) – e(1) = ½ (k2 + k2 + .......)
From degree of v1 From degree of v2 Other edges
= ½ (2k2 + .......) = k2 + ½ (.......)
How to find the BI of one particular inner cycle labeling
v1v2 v3 v4 v5
0 0 1 1 1
|k2 + 0 - k3 - k4 + 0|
C3 with even vertices
v1 v2 v3 Balance Index
0 0 0 |k1 + k2 + k3|
0 0 1 |k1|
0 1 1 |-k2|
0 1 0 |k3|
Odd number of vertices Total vertices =
Difference between 2M and 2M+1 graphs:• Extra vertex in 2M+1 ends up in outer cycle (degree 2)• Extra vertex can be labeled 1-vertex or 0-vertex• Corollary implies that this will ±1 to each member of 2M’s set