Affiliation:
1. Department of Mathematics, Pennsylvania State University at Abington , Abington , PA, 19001 , USA
Abstract
Abstract
Let
G
=
(
V
(
G
)
,
E
(
G
)
)
G=\left(V\left(G),E\left(G))
be a graph of order
n
n
. The exponential atom-bond connectivity matrix
A
e
ABC
(
G
)
{A}_{{e}^{{\rm{ABC}}}}\left(G)
of
G
G
is an
n
×
n
n\times n
matrix whose
(
i
,
j
)
\left(i,j)
-entry is equal to
e
d
(
v
i
)
+
d
(
v
j
)
−
2
d
(
v
i
)
d
(
v
j
)
{e}^{\sqrt{\tfrac{d\left({v}_{i})+d\left({v}_{j})-2}{d\left({v}_{i})d\left({v}_{j})}}}
if
v
i
v
j
∈
E
(
G
)
{v}_{i}{v}_{j}\in E\left(G)
, and 0 otherwise. The exponential atom-bond connectivity energy of
G
G
is the sum of the absolute values of all eigenvalues of the matrix
A
e
ABC
(
G
)
{A}_{{e}^{{\rm{ABC}}}}\left(G)
. It is proved that among all trees of order
n
n
, the star
S
n
{S}_{n}
is the unique tree with the minimum exponential atom-bond connectivity energy.
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