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=
V
1
,
V
2
,...,
V
n
be a col
le
ction of
n
V
Definition 2.26
(Zhao et al. 2012a) Let
=
V
R
r
ij
)
n
×
n
an intuitionistic f
u
zzy
re
lation over
V
×
V
. Then
G
nodes, and
R
=
(
,
=
E
k
=
V
i
V
j
|∀
V
i
,
V
j
∈
V
, then
V
E
is called an intuitionistic fuzzy graph. If
E
,
is called a basic graph of
V
R
.
,
A path in a graph is a sequence of edges joining two nodes as (
ABCD
)
. A circuit
is a closed path as (
ABCA
. A connected graph has paths between any pair of nodes.
A tree is a connected graph with no circuits and a spanning tree of a connected graph
)
isatreeingraph
V
R
(Zahn 1971).
If we add every edge a weight and define the weight of a tree to be the sum of the
weights of its constituent edges, then
R
which contains all nodes of
V
,
,
Definition 2.27
(Zahn 1971) A minimum (maximum) spanning tree of a graph
V
R
is a spanning tree whose weight is minimum (maximum) among all span-
ning trees of the graph
V
,
R
.
We usually compute the minimum (maximum) spanning tree of a graph
V
,
R
by Kruskal method (Kruskal 1956) or Prim method (Prim 1957). Because of the
complexity of the objective world and the fuzziness of the human perception, the data
information needed to be clustered is often imprecise or uncertain and sometimes
is given by IFSs. In such situations, some effective and convenient intuitionistic
clustering algorithms are needed. The MST (minimum spanning tree) clustering
algorithm was first proposed by Zahn (1971), whose basic idea is that: a multi-
attribute sample point can be considered as a point of a multi-dimensional space.
If the density of the sample points in some regions in the multi-dimensional space
is high, while in other regions is low or even blank, then the high-density regions
can be separated from the blank or the low-density regions naturally, so that we get
the clustering structure of the sample points which best embodies the distribution of
the sample points. Based on the idea of Zahn (1971), Zhao et al. (2012a) introduced
an intuitionistic fuzzy clustering method called intuitionistic fuzzy MST clustering
algorithmbased on the graph theoretic techniques and the intuitionistic fuzzy distance
measure to cluster intuitionistic fuzzy information. In the following, we first introduce
the concepts of intuitionistic fuzzy distance measure and intuitionistic fuzzy distance
matrix:
,
Definition 2.28
(Zhao et al. 2012a) Let
A
j
(
j
=
1
,
2
,...,
n
)
be
n
IFSs. Then
D
=
d
ij
n
×
n
d
A
i
,
A
j
=
is called an intuitionistic fuzzy distance matrix, where
d
ij
=
−
ϑ(
1
is the intuitionistic fuzzy distance between
A
i
and
A
j
, which has the
following properties:
A
1
,
A
2
)
(1)
d
ij
(
i
,
j
=
1
,
2
,...,
n
)
are IFVs.
(2)
d
ij
=
(
0
,
1
)
if and only if
A
i
=
A
j
.
(3)
d
ij
=
d
ji
, for all
i
,
j
=
1
,
2
,...,
n
,
where
ϑ(
A
1
,
A
2
)
is defined in Theorem 2.2.
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