On extremal cacti with respect to the first degree-based entropy

Abstract

Let GG be a simple graph with degree sequence D(G)=(d1,d2,…,dn)D\left(G)=\left({d}_{1},{d}_{2},\ldots ,{d}_{n}). The first degree-based entropy of GG is defined as I1(G)=ln∑i=1ndi−1∑i=1ndi∑i=1n(dilndi){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 nn vertices and kk cycles.

Keywords

• graph entropy
• cactus
• first degree-based entropy
• 05c50