BALANCE INDEX SETS OF BI-DEGREE AND TRI-DEGREE

GRAPHS

Terry RoseProfessor Hsin-hao Su

Background Graph

Vertices: V(G) and v(n)Edges: E(G) and e(n)UndirectedSimple

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

Balance Index Sets

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}

How exactly to calculate BI Every vertices combination

Place 0’s first then remaining vertices are 1’s

Balance Remove redundant balance indexes

ExampleTracker for set [0, 1, 2]0 appeared 6 times1 appeared 12 times2 appeared 2 times

Tracker for set [0, 1, 2]0 appeared 3 times1 appeared 6 times2 appeared 1 times

Graph Theory

Cycle graph

n vertices in a circleInner vertices

Star graphm vertices to each inner vertexOuter vertices

Number of vertices = m*n+n

First Case m is oddNumber of vertices is even

Second Case m is evenNumber of vertices is odd

composes an inner cycle to each vertex of

Composed of two disjoint sets, A and B |A| = m and |B| = n

Every vertex in A has an edge to every vertex in B

Any Graph with bi-degree vertex set

|A|+|B| is even Let there be m 0-vertices with deg a

M-m 0-vertices with deg b|A|-m 1-vertices with deg aM-(|A|-m) 1-vertices with deg bFor purposes of generality, |A|<|B|

|A|+|B| is odd Floater vertex placed in set B

Bi-degree vertex set plus one vertex