Alternative Proofs on the Indices of Cacti and Unicyclic Graphs with $n$ Vertices

Sudipta Mallik

Let $H_n$ be the cactus obtained from the star $K_{1,n-1}$ by adding $\lfloor \frac{n-1}{2}\rfloor$ independent edges between pairs of pendant vertices. Let $K_{1,n-1}^{+}$ be the un…