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

1. Terry RoseProfessor Hsin-hao Su balance Index Sets of Bi-degree and Tri-degree Graphs

2. Background • Graph • Vertices: V(G) and v(n) • Edges: E(G) and e(n) • Undirected • Simple • |v(0)-v(1)| • |e(0)-e(1)| • Label

3. Balance Index Sets

4. 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)|= 1 BI(G) = {0,1,2}

5. How exactly to calculate BI • Every vertices combination • Place 0’s first then remaining vertices are 1’s • Balance • Remove redundant balance indexes

6. 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 times

7. Graph Theory

8. Cycle graph • n vertices in a circle • Inner vertices • Star graph • m vertices to each inner vertex • Outer vertices • Number of vertices = m*n+n

10. First Case m is odd • Number of vertices is even

11. Second Case m is even • Number of vertices is odd

12. composes an inner cycle • to each vertex of

13. 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

14. Any Graph with bi-degree vertex set

15. |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 a • M-(|A|-m) 1-vertices with deg b • For purposes of generality, |A|<|B|

16. |A|+|B| is odd • Floater vertex placed in set B

20. Bi-degree vertex set plus one vertex