International Journal of Scientific and Innovative Mathematical Research (IJSIMR)
Volume 3, Issue 12, December 2015, PP 9-12
ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)
www.arcjournals.org
©ARC Page | 9
A Note on Semitotal blocks in Fuzzy Graphs
Shaik Mohiddin Shaw
Department of Basic Sciences and Humanities,
Narasaraopeta Engineering College, Narasaraopet, A.P., India.
mohiddin_shaw@yahoo.co.in
Pulivarthy Masthan
Department of Mathematics
Krishna University, Machilipatnam, A.P.., India
mastan.chinna@gmail.com
Ch. Baby Rani
Department of Mathematics,
V.R. Siddhartha Engineering College, Vijayawada, A.P., India.
ranicbrl@gmail.com
Abstract: In this research paper, it was studied about degree of vertices of a semitotal blocks in fuzzy graphs.
In process we obtain some interesting results regarding the degree of the vertices in semitotal blocks in fuzzy
graphs. We observed that when ‘B’ is a block of a given fuzzy graph G:(V, σ, µ), then degree of the vertex B in
semi total block fuzzy graph T
STB
F(G) is equal to the sum of the membership grade of the vertices in that block
and the number of edges in T
STB
F(G) related to block B is V(B) with membership grade minimum of σ(u),σ(B).
Also, we obtained that When G:(V , σ, μ ) fuzzy graph and v be a fuzzy vertex with degree d
FG
(v) in G:(V , σ, μ ), then
the degree of ‘v’ in semitotal block fuzzy graph T
STBF
(G), d
STFG
(v) equal to the sum of the degree of the vertex in
fuzzy graph and the product of {B/Bis a block in fuzzy graph containing v} with minimum of the set {σ(v), σ(B)}.
Finally, it is proved that the ring sum of given fuzzy graph and vertex block fuzzy graph is equals to the
semitotal block fuzzy graph of given fuzzy graph.
Keywords: fuzzy graph, ring sum of fuzzy graphs, Degree of vertex in fuzzy graphs, Semitotal-block fuzzy
graph, vertex block graph.
1 INTRODUCTION
Rosenfeld considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975.
Fuzzy models playing a vital role in real situations and becoming useful in engineering and sciences.
In this paper, we introduced an algebraic operation ring sum of two fuzzy graphs which is different
from existed literature and provided necessary examples. In this connection it is observed that the ring
sum of two fuzzy graphs is also a fuzzy graph. Several interesting results on semitotal block fuzzy
graph T
STB
F(G) of a fuzzy graph are observed. Finally, it is proved that, When G:(V ,σ, μ) fuzzy graph
and v be a fuzzy vertex with degree d
FG
(v) in G:(V,σ,μ ), then the degree of „v‟ in semitotal block
fuzzy graph T
STBF
(G) is d
STFG
(v) equal to the sum of the degree of the vertex in fuzzy graph and the
product of {B/Bis a block in fuzzy graph containing v} with min { σ(v) and σ(B)}. Also we proved
that T
STBF
(G) = G:(V ,σ,μ) B
v
(STBF(G)).
Throughout this paper, we assume that fuzzy graph G is simple and connected.
1.1 Definition : A connected non–trivial fuzzy graph having no fuzzy cut vertex is a block in fuzzy
graph.
1.2 Note: The set of all Blocks of fuzzy graph G is denoted by SBF(G)
1.3 Example: Consider the following fuzzy graph G:(V, σ, µ). Where V = {V
1
, V
2
, V
3
, V
4
, V
5
, V
6
,
V
7
, V
8
, V
9
, V
10
, V
11
} with σ(v
1
) = 0.8, σ(v
2
) = 1, σ(v
3
) = 0.7, σ(v
4
) = 0.8, σ(v
5
) = 0.8, σ(v
6
) = 0.8,
σ(v
7
) = 0.8, σ(v
8
) = 0.8, σ(v
9
) = 0.7, σ(v
10
) = 0.8, σ(v
11
) = 0.8 and μ(v
1
, v
2
) = 0.4, μ(v
2
, v
3
) = 0.2, μ(v
1
,