Abstract
A bond incident degree (BID) index of a graph
G
is defined as
∑
u
v
∈
E
G
f
d
G
u
,
d
G
v
, where
d
G
w
denotes the degree of a vertex
w
of
G
,
E
G
is the edge set of
G
, and
f
is a real-valued symmetric function. The choice
f
d
G
u
,
d
G
v
=
a
d
G
u
+
a
d
G
v
in the aforementioned formula gives the variable sum exdeg index
SEI
a
, where
a
≠
1
is any positive real number. A cut vertex of a graph
G
is a vertex whose removal results in a graph with more components than
G
has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by
V
n
,
k
the class of all
n
-vertex graphs with
k
≥
1
cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of
SEI
a
from the set
V
n
k
for
a
>
1
. In the present paper, we not only characterize the graphs with the minimum value of
SEI
a
from the set
V
n
k
for
a
>
1
, but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all
n
-vertex molecular graphs with
k
≥
1
cut vertices and containing at least one cycle.