1. Dan Bouchard, Patrick Clark*, Hsin- Hao Su Stonehill College

2. 1 1 0 0 0 x 1 x 0 0 Let G be a simple graph with vertex set V (G) and edge set E(G). Let Z 2 = {0, 1}. An edge labeling f : E(G) → Z 2 induces a vertex partial labeling f + : V (G) → Z 2 defined by f + (v) = 0 if the edges labeled 0 incident on v is more than the number of edges labeled 1 incident on v, and f + (v) = 1 if the edges labeled 1 incident on v is more than the number of edges labeled 0 incident on v.

3.  e f (0) = # of 0-edges e f (1) = # of 1-edges v f (0) = # of 0-vertices v f (1) = # of 1-vertices  e f (0) = 5 e f (1) = 4 v f (0) = 5 v f (1) = 2 1 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 An edge labeling f of a graph G is said to be edge-friendly if | e f (0) − e f (1) |≤ 1. A graph G is said to be an edge- balanced graph if there is an edge-friendly labeling f of G satisfying | v f (0) − v f (1) | ≤ 1.

4. The edge-balance index set of the graph G, EBI(G), is defined as {| v f (0) − v f (1) | : the edge labeling f is edge-friendly.}.

5.  An L-product with cycle by star graph is composed of a cycle graph and n star graphs. C 3 St(2) C 3 X L St(2)

6.  In particular, the focus of our research consisted of analyzing graphs of the form C n X L St(m), where m is even and greater than 2.

7.  In a C n X L St(m) m is even and greater than 2, we notice that all of the vertices are grouped into n packages. These packages consist of m degree 1 vertices and 1 degree m+2 vertex. Package Each of the m degree 1 vertices must be labeled, and it's labeling is completely dependent on the labeling of it's incident edge. The degree m + 2 vertex in each package ho can be either labeled or unlabeled.

8.  Remember that for every friendly labeling, the EBI Is given by | v f (0) − v f (1) |.  For the purposes of my study, it is essential to determine the highest EBI possible for a particular graph.  Two main strategies seem apparent, either maximize v f (0) or minimize v f (1).

9. 0 C 3 X L St(4) 0 0 0 0

10. 1 1 1 x

12. Step 1 - Label each of the n edges of the cycle with a 1. Step 2 - For as many packages as possible given the amount of 0-edgeswe have, label one more than half of the edges incident to a degree 1 vertex in a package with a 0. Step 3 - Starting at the package next to the last package we were able to label by Step 2, label each degree 1 edge with a 1 until we have used all of our 0-edges. Step 4 - Label the rest of our edges with 1. 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1

15.  After developing the algorithm to produce the highest EBI for our graphs, we considered how to lower the EBI for our graphs to produce and EBI set.  We proceeded by considering how we can make switches within our highest EBI labeling that can effectively lower our EBI by 1 or 2.

16. Switch!!!!