Extremal Chemical Graphs for the Arithmetic-Geometric Index

TL;DR

This work addresses maximizing the arithmetic-geometric index $AG(G)$ for chemical graphs of fixed order $n$ and size $m$ by deriving a sharp upper bound and a complete extremal characterization. The authors develop a framework using edge-degree costs, degree-count vectors, and a quadruple-based function $f$ to express $AG(G)$ and identify a constructive family $\mathcal{G}_{n,m}$ whose graphs attain the bound for all nonexceptional pairs $(n,m)$. They also prove that removing the connectivity constraint does not yield a larger $AG$, and they isolate 22 exceptional pairs with explicit extremal graphs $H_{n,m}$. The results yield a practical, closed-form bound $UB_{n,m}$ dependent on $n$, $m$, and the congruence class of $2m-n$ modulo $3$, and provide a comprehensive picture of extremal chemical trees, unicyclic, and bicyclic graphs, with implications for the study of degree-based topological indices in chemical graph theory.

Abstract

The arithmetic-geometric index is a newly proposed degree-based graph invariant in mathematical chemistry. We give a sharp upper bound on the value of this invariant for connected chemical graphs of given order and size and characterize the connected chemical graphs that reach the bound. We also prove that the removal of the constraint that extremal chemical graphs must be connected does not allow to increase the upper bound.

Figures (4)

• Figure 1: Examples of graphs in $\mathcal{G}_{n,m}$ for some pairs $(n,m)$.
• Figure 2: Seven connected extremal chemical graphs with all edges having at least one endpoint of degree 4 and with $n_0(G)>0$ or $n_2(G)+n_3(G)>1$.
• Figure 3: Fifteen connected extremal chemical graphs with at least one edge having no endpoint of degree 4.
• Figure 4: Illustration of the algorithm in the proof of Theorem \ref{['thm1']}

Theorems & Definitions (28)

• Definition 1
• Corollary 1
• Corollary 2
• proof
• proof
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof