Sharp bounds on the symmetric division deg index of graphs and line graphs

TL;DR

This paper establishes precise bounds for the symmetric division degree index in graphs and line graphs, highlighting its mathematical properties and extremal cases, which are relevant for chemical graph theory.

Contribution

It provides sharp bounds and characterizations for the symmetric division degree index of graphs and their line graphs, advancing understanding of this index's properties.

Findings

Sharp bounds on the SDD index for graphs and line graphs

Characterization of extremal graphs achieving bounds

Enhanced understanding of the SDD index's mathematical properties

Abstract

For a graph $G$ with vertex set $V_{G}$ and edge set $E_{G}$, the symmetric division deg index is defined as $SDD(G)=\sum\limits_{uv\in E_{G}}(\frac{d_{u}}{d_{v}}+\frac{d_{v}}{d_{u}})$, where $d_{u}$ denotes the degree of vertex $u$ in $G$. In 2018, Furtula et al. confirmed the quality of SDD index exceeds that of some more popular VDB indices, in particular that of the GA index. They shown a close connection between the SDD index and the earlier well-established GA index. Thus it is meaningful and important to consider the chemical and mathematical properties of the SDD index. In this paper, we determine some sharp bounds on the symmetric division deg index of graphs and line graphs and characterize the corresponding extremal graphs.