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