balance Index - PowerPoint Presentation

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balance Index Sets of Bi-degree and Tri-degree Graphs

Terry Rose

Professor Hsin-hao SuSlide2

BackgroundGraphVertices: V(G) and v(n)Edges: E(G) and e(n)

Undirected

Simple

|v(0)-v(1)||e(0)-e(1)|LabelSlide3

Balance Index SetsSlide4

Example of BI

v(0) = 3 v(1) = 3

|v(0)-v(1)| = 0 ≤1

e(0) = 2 e(1) = 1

|e(0)-e(1)|= 1BI(G) = {0,1,2}Slide5

How exactly to calculate BIEvery vertices combinationPlace 0’s first then remaining vertices are 1’s

Balance

Remove redundant balance indexesSlide6

Example

Tracker for set [0, 1, 2]

0 appeared 6 times

1 appeared 12 times

2 appeared 2 times

Tracker for set [0, 1, 2]

0 appeared 3 times

1 appeared 6 times

2 appeared 1 timesSlide7

Graph Theory Slide8

Cycle graphn vertices in a circleInner verticesStar graph

m vertices to each inner vertex

Outer vertices

Number of vertices = m*n+nSlide9
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First Case m is odd

Number of vertices is evenSlide11

Second Case m is even

Number of vertices is oddSlide12

composes an inner cycle to each vertex of Slide13

Composed of two disjoint sets, A and B |A| = m and |B| = nEvery vertex in A has an edge to every vertex in B Slide14

Any Graph with bi-degree vertex setSlide15

|A|+|B| is evenLet there be m 0-vertices with deg aM-m 0-vertices with deg b

|A|-m 1-vertices with deg a

M-(|A|-m) 1-vertices with deg

bFor purposes of generality, |A|<|B|Slide16

|A|+|B| is oddFloater vertex placed in set BSlide17
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Bi-degree vertex set plus

one vertex