Inequalities between degree- and distance-based graph invariants

Imran Nadeem,Hani Shaker

doi:10.1186/s13660-018-1633-y

Imran Nadeem, Hani Shaker

Open Access

https://doi.org/10.1186/s13660-018-1633-y

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Abstract

Inequalities provide a way to study topological indices relatively. There are two major classes of topological indices: degree-based and distance-based indices. In this paper we provide a relative study of these classes and derive inequalities between degree-based indices such as Randić connectivity, GA, ABC, and harmonic indices and distance-based indices such as eccentric connectivity, connective eccentric, augmented eccentric connectivity, Wiener, and third ABC indices.

Highlights

Let G = (VG, EG) be a simple connected graph with a vertex set VG and an edge set EG
We present the inequality relations of the harmonic index with eccentric connectivity and connective eccentric indices for self-centered graphs
Some inequality relations have been studied between two topological indices belonging to degree-based and distance-based indices

Summary

Introduction

3.3 Harmonic index in relation with distance-based indices In the following theorem, we derive the inequalities between the harmonic index and certain distance-based indices such as eccentric connectivity, connective eccentric, augmented eccentric connectivity, and Wiener indices for any non-self-centered graph. We present the inequality relations of the harmonic index with eccentric connectivity and connective eccentric indices for self-centered graphs. We establish an inequality between harmonic and augmented eccentric connectivity indices for regular self-centered graphs. From the compound inequality (8), we obtain the equality relation of the diameter of G with the augmented eccentric connectivity indices as follows: ndd dG = ξ ac(G) By using this relation in inequality (26), we get the required result.

It can be written as

Conclusion