On Total Vertex Irregularity Strength of Hypercubes
The concept of total vertex irregularity labelling was introduced by Ba?ca et al. (2007). Let G = (V;E) be a simple graph (without loop and multiple edges). A function : V [ E ! f1; 2; :::; kg is called a vertex-irregular total k????labelling of G if for any two different vertices u and v in V s...
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| Format: | Theses |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/45933 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | The concept of total vertex irregularity labelling was introduced by Ba?ca et al.
(2007). Let G = (V;E) be a simple graph (without loop and multiple edges).
A function : V [ E ! f1; 2; :::; kg is called a vertex-irregular total k????labelling
of G if for any two different vertices u and v in V satisfy wt(u) 6= wt(v), where
wt(u) = (u)+
P
uw2E (uw): The total vertex irregularity strength of G, denoted
by tvs(G), is the smallest positive integer k for which G has a vertex irregular total
k-labelling.
Baca, et al. (2007) derived the lower and upper bounds for any r-regular graph
G with p vertices and q edges as follows:
p+r
r+1
tvs(G) p ???? r + 1: In addition,
Nurdin, et al. (2010) also give a conjecture for a connected graph G that is
tvs(G) = max
n+n
+1
;
l
+n+n+1
+2
m
; :::;
l
+
P
i= ni
+1
mo
, where ni is the number of
vertices of degree i = ; +1; +2; :::;, where and are the minimum and the
maximum degree of G, respectively.
In this thesis, we determine the total vertex irregularity strength of hypercube graph
with 2n vertices namely, tvs(Qn) =
2n+n
n+1
for n 14. This result achieves the
lower bound given by Ba?ca, et al. (2007) and also strengthens the conjecture by
Nurdin, et al.(2010). |
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