Transcript On the Edge Balance Index of Flux Capacitor Graphs, L

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 Z2 = {0, 1}.
• An edge labeling f : E(G) → Z2 induces a vertex
partial labeling f+ : V (G) → Z2 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 edgefriendly if |ef (0) − ef (1)|≤ 1. A graph G is said to be
an edge-balanced graph if there is an edge-friendly
labeling f of G satisfying |vf (0) − vf (1)| ≤ 1.
The edge-balance index set of the graph G, EBI(G),
is defined as {|vf (0) − vf (1)| : the edge labeling f is
edge-friendly.}.
Centipede Graphs
• A Centipede Graph is an LProduct with Cycle by
Cycle graph composed of
two cycle graphs. A cycle,
Cn, consists of n vertices
each connected to 2
others to form a cycle
where n≥3. There is then a
cycle, Cm, attached to each
vertex of Cn. 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, Cm, 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, Cn, 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 Cn. It is
represented as CnXLSt(m).
EBI(CnXLSt(m)) Theorem
How CnXLSt(m) is labeled
• First, the n edges of
the Cn 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(C3XLSt(2))={0,1, 2,3}
EBI(C3XLSt(3))={0,1,2,3,4}
EBI(C3XLSt(4))={0,1, 2,3,4,5}