On the Edge Balance Index of Centipede Graphs and L-Product of Cycle by Star Graphs Meghan Galiardi, Daniel Perry*, Hsin-Hao Su Stonehill College

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. f + (v) is not defined if the number of edges labeled by 0 is equal to the number of edges labeled by 1.

Definitions 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.

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.}.

Centipede Graphs A Centipede Graph is an L- Product with Cycle by Cycle graph composed of two cycle graphs. A cycle, C n, consists of n vertices each connected to 2 others to form a cycle where n≥3. There is then a cycle, C m, attached to each vertex of C n. Ce(n,m) is the abbreviation.

EBI(Ce(n,m)) Theorem

How Centipede Graphs are Labeled We begin by labeling as many edges adjacent to degree 4 vertices by 1. The rest of the edges are then labeled by 0. This process ensures that any outer cycle, C m, with a 1-vertex of degree 4 will not be composed completely of 1-vertices and that any outer cycle with a 0-vertex of degree 4 will have m 0-vertices. If a degree 4 vertex is left unlabeled, the cycle will still consist of 0-vertices and no 1-vertices.

EBI(Ce(3,3))={0,1}

EBI(Ce(4,3))={0,1, 2}

EBI(Ce(5,3))={0,1, 2,3}

EBI(Ce(3,4))={0,1, 2,3}

EBI(Ce(7,4))={0,1, 2,3,4,5,6}

EBI(Ce(4,5))={0,1, 2,3,4}

L-Product with Cycle by Star Graphs An L-product with cycle by star graph is composed of a cycle graph and n star graphs. A cycle, C n, consists of n vertices each connected to 2 others to form a cycle where n≥3. There is then a Star graph, St(m), attached to each vertex of C n. It is represented as C n X L St(m).

EBI(C n X L St(m)) Theorem

How C n X L St(m) is labeled First, the n edges of the C n are all labeled by 1’s. The remaining 1-edges are then used on the outer St(m)’s, filling one star before moving on to the next. The remaining edges are all labeled by 0, which will yield the highest possible EBI.

EBI(C 3 X L St(2))={0,1, 2,3}

EBI(C 3 X L St(3))={0,1,2,3,4}

EBI(C 3 X L St(4))={0,1, 2,3,4,5}