Abstract
The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph
G
is defined as
∑
v
w
∈
E
G
d
v
+
γ
d
w
+
γ
−
1
/
2
, where
γ
is a non-negative real number and
d
w
is the degree of a vertex
w
in
G
. We call this index as the variable Randić index and denote it by
R
v
γ
. In this paper, we show that the graph created from the star graph of order
n
by adding an edge has the minimum
R
v
γ
value among all unicyclic graphs of a fixed order
n
, for every
n
≥
4
and
γ
≥
0
.