Extremal Topologies for the Merrifield-Simmons Index on Dynamic Trees

Abstract

In this article, we study the recognition of extremal topologies for the Merrifield-Simmons index in the space of tree graphs. We analyze how to obtain the maximum and the minimum number of independent set on these topologies when a new vertex v is joined to a tree $T_n$ via a new edge $\{v_p,v\}$, with $v_p \in V(T_n)$ and $v \notin V(T_n)$.

We show that $i(T_n \cup \{ \{v_p, v\} \})$ is minimum when v is a new leaf node, and its father $v_p$ was also a leaf node in $T_n$. In addition, the father $v_h$ of $v_p$ has a maximal degree in $T_n$, and as a last criterion, $v_p$ has a maximal eccentricity into the nodes in $T_n$ with maximal degree. On the other hand, we show that $i(T \cup \{ \{v_p, v\} \})$ is maximum when v is linked to a vertex $v_p$ with maximal degree in $T_n$, and $v_p$ has a greater number of neighbors with minimal degree in $T_n$.

In memory of Miguel Rodríguez.