Affiliation:
1. School of Mathematics and Statistics, Guangdong University of Technology , Guangzhou 510090 , P. R. China
2. School of Mathematical Sciences, South China Normal University , Guangzhou 510631 , P. R. China
Abstract
Abstract
Let
G
G
be a simple graph with degree sequence
D
(
G
)
=
(
d
1
,
d
2
,
…
,
d
n
)
D\left(G)=\left({d}_{1},{d}_{2},\ldots ,{d}_{n})
. The first degree-based entropy of
G
G
is defined as
I
1
(
G
)
=
ln
∑
i
=
1
n
d
i
−
1
∑
i
=
1
n
d
i
∑
i
=
1
n
(
d
i
ln
d
i
)
{I}_{1}\left(G)=\mathrm{ln}{\sum }_{i=1}^{n}{d}_{i}-\frac{1}{{\sum }_{i=1}^{n}{d}_{i}}{\sum }_{i=1}^{n}\left({d}_{i}\mathrm{ln}{d}_{i})
. In this article, we give sharp upper and lower bounds for the first degree-based entropy of graphs in
C
(
n
,
k
)
{\mathcal{C}}\left(n,k)
and characterize the corresponding extremal graphs when each bound is attained, where
C
(
n
,
k
)
{\mathcal{C}}\left(n,k)
is the set of all cacti with
n
n
vertices and
k
k
cycles.
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