On the Edge Balance Index of the L-Product of Cycle by Star - PowerPoint Presentation

Slide1

On the Edge Balance Index of the L-Product of Cycle by Star Graphs

Dan Bouchard, Patrick Clark*,

Hsin-Hao

Su

Stonehill

CollegeSlide2

Introduction

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

. Slide3

Notation

e

f

(0

) = # of 0-edges ef (1

) = # of 1-edges vf (0) = # of 0-vertices

vf (1) = # of 1-verticese

f (0) = 5 ef

(1) = 4 vf (0)

= 5 vf (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.Slide4

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.}.Slide5

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.

C

3

St(2)

St(2)

St(2)

C

3

X

L

St(2)Slide6

Focus

In particular, the focus of our research consisted of analyzing graphs of the form

C

nX

LSt(m), where m is even and greater than 2.Slide7

Basis Structure of Graph

In a

C

nXL

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

E

ach of the m degree 1 vertices must be labeled, andit'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.

Slide8

Labeling Approach

Remember that for every friendly labeling, the EBI Is given by |

v

f (0) −

vf (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

vf (0) or minimize vf (1).Slide9

Maximize v

f

(0)

0

Although a similar argument can be made otherwise, lets consider a graph in which the total number of vertices is even.

Number of 0-edges:

, where n is the

n

umber of vertices accounted for by the cycle, and m is the number of outer vertices for each star.

We have m+2 edges in each package, and to maximize

v

f

(0

),

we must label in such a way to make as many of the red vertices as possible 0-vertices.

This requires that

:

+

1 edges in a package be 0-edges, and we place these on edges outside of the cycle, as each vertex out side the cycle is only degree 1..

By doing this for as many packages as possible before we run out of 0-edges to work with, we will maximize

v

f

(0)

C

3

X

L

St(4)

0

0

0

0Slide10

Minimize v

f

(1)

1

Number

of

1-edges

:

, where n is the

n

umber of vertices accounted for by the cycle, and m is the number of outer vertices for each star.

We have m+2 edges in each package, and to minimize

v

f

(1),

we must label in such a way to make as many of the red vertices as possible unlabeled.

This requires that

:

edges in a package be 1-edges.

By doing this for as many packages as possible before we run out of 1-edges to work with, we will minimize

v

f

(1)

C

3

X

L

St(4)

1

1

xSlide11

Analysis

For the case of maximizing

v

f

(0), using the division algorithm we can write :

=(

) k +

r, where k is the number of packages where we were able to create a 0-vertex on the cycle, and r is the number of 0-edges that left to be placed on the outer edges.For the case of minimizing vf

(1), using the division algorithm we can write :

=(

)

q + j, where j

is the number of packages where we were able to create an unlabeled vertex on the cycle, and r is the number of 0-edges that left to be placed on the outer edges.By using algebra to compare these equations, we arrive at the result that k>q, allowing us to conclude that it is more effective to maximize

vf (0)

 Slide12

Labeling Strategy

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

1Slide13

Highest EBI Theorem

Given our equations relating the total number of 0-edges and 1-edges to the number required to make the degree m+2 vertex in a package the same label, we were able to conclude the following

Equation 1:

=(

) k + r

.

Equation 2:

=(

)

q + jTheorem: The Highest Edge Balance Index for any Cn

xL St(m) graphs where m is even and greater than 2 is

{2k+1 if n is odd and r <

} {2k+2 if n is odd and r >=

}

{2q if n is even and j <

} {

2q+1 if n is even and j >=

}

 Slide14
Slide15

EBI Sets

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.Slide16

Lemma

If a packages contains at least

+1 0-edges and

at

least 1 1-edge,

we can make a label switch within that package to reduce the EBI by 2.

Switch!!!!Slide17
Slide18

Lemma

For any friendly labeling that contains at least 1 package that is

composed of less than

-1 0-edges

or 1-edges and at least 1

package that is composed of exactly

+1 edges of the same label, we can make a label switch to increase our EBI by 1.

..... switch!!!..

.

 Slide19
Slide20

Theorem

Theorem: The Highest Edge Balance Index for any

Cn

xL St(m) graphs where m is even and greater than 2 is {0,1,2,…, 2k+1 if n is odd and r <

} {0,1,2,…, 2k+2 if n is odd and r >=

}

{0,1,2,…, 2q if n is even and j <

} {0,1,2,…, 2q+1 if n is even and j >=

}

 Slide21