balance index sets of bi-degree and tri-degree graphs
Post on 16-Feb-2016
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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
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