1. Balance Index Set of Generalized Ear Expansion Hsin-hao Su Patrick Clark Dan Bouchard*

2. Labeling the graph Let G be a simple graph with vertex set V(G) and edge set E(G) and let Z 2 = {0,1}. : 0 vertex : 1 vertex I A labeling f: V(G)  Z 2 which induces an edge partial labeling f * : E(G)  Z 2 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 |v f (0) - v f (1)| ≤ 1 The BI(G), the balance index of G, is defined as: {|e f (0) - e f (1)| : the vertex labeling f is friendly.} BI(G) = |1 – 1| = 0

3. Finding a Balance Index (BI) : 0 vertex : 1 vertex I Create a friendly labeling  Difference between 0 and 1 vertices less than or equal to 1 Five 0-vertices Six 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-edges Four 1-edges BI =|2-4| = 2

4. Generalized Ear Expansion 1 2 3 4 5 k 2 = 2 k 1 = 3 k 3 = 2 k 5 = 2 k i = Number of ear expansions on the corresponding edge i

5. Algebraic Equalities Adapted from Kwong and Shiu

6. Even number of vertices

7. 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

8. 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 labeling of the inner cycle’s vertices is also important

9. A closer look at a single edge of the inner cycle v1v1 v2v2 1 2 k 2 = 2 Possibility 1: v 1 and v 2 are both 0-vertices e(0) – e(1) = ½ (k 2 + k 2 +.......) From degree of v 1 From degree of v 2 Other edges = ½ (2k 2 +.......) = k 2 + ½ (.......)

10. How to find the BI of one particular inner cycle labeling v1v1 v2v2 v3v3 v4v4 v5v5 00111 |k 2 + 0- k 3 - k 4 + 0|

11. C 3 with even vertices v1v1 v2v2 v3v3 Balance Index 000|k 1 + k 2 + k 3 | 001|k 1 | 011|-k 2 | 010|k 3 |

12. 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