Abstract
Let \(G\) be a connected graph of order \(n\) with degree sequence
\(D(G) = [d_1, d_2, \ldots, d_n]\). The first degree-based entropy
of \(G\) is defined as
\[
I_1(G) = \ln\left(\sum_{i=1}^n d_i\right) -
\frac{1}{\sum_{i=1}^n d_i} \sum_{i=1}^n (d_i\ln d_i)
\]
In this paper, we characterize the corresponding extremal graphs
which attain the maximum value of \(I_1(G)\) among all \(k\)-cyclic
graphs of order \(n\), where \(k \geq 1\).