[Paper Review] Novel Zagreb Indices-Based Inequalities with Particular Regard to Semiregular and Generalized Semiregular Graphs

Tamás Réti, Imre Felde|arXiv (Cornell University)|Sep 26, 2015

Graph theory and applications37 references6 citations

TL;DR

This paper introduces novel inequalities involving three degree-based topological indices—M₁(G), M₂(G), and F(G)—for connected graphs, demonstrating that equality holds precisely when the graph is regular or semiregular. It further proposes weakly semiregular graphs as a generalization of traditional bidegreed semiregular graphs and applies Zagreb index-based irregularity measures to evaluate fullerene isomers, reevaluating graph irregularity for stability prediction in nanomaterials.

ABSTRACT

Topological relations between three degree-based invariants of a connected graph G are investigated. We present novel inequalities including M1(G), M2(G) and F(G), and show that in all cases equality holds if G is a regular or a semiregular graph. Additionally, the notion of so called weakly semiregular graphs is introduced, they are considered as a possible generalization of traditional bidegreed semiregular graphs. Based on the use of Zagreb indices based graph irregularity indices, for purposes of fullerene stability prediction, comparative tests have been performed on a finite set of dual graphs of C40 fullerene isomers. By using the findings obtained, the traditional concept of graph irregularity characterization has been critically reevaluated.

Motivation & Objective

• To establish new inequalities linking the first and second Zagreb indices (M₁, M₂) and the F-index (F) in connected graphs.
• To investigate the conditions under which equality holds in these inequalities, particularly in regular and semiregular graphs.
• To introduce and analyze the concept of weakly semiregular graphs as a generalization of traditional bidegreed semiregular graphs.
• To assess the applicability of Zagreb index-based irregularity measures in predicting the stability of C40 fullerene isomers through comparative analysis of dual graphs.
• To critically reevaluate the traditional characterization of graph irregularity using topological indices in chemical graph theory.

Proposed method

• Derives novel analytical inequalities connecting M₁(G), M₂(G), and F(G) using degree sequences of connected graphs.
• Introduces the concept of weakly semiregular graphs, defined by degree sets with at most two distinct values but without requiring uniformity across partitions.
• Applies Zagreb index-based irregularity indices to a finite set of dual graphs derived from C40 fullerene isomers.
• Performs comparative testing of these indices to evaluate their effectiveness in distinguishing structural stability.
• Uses mathematical analysis and graph-theoretic reasoning to prove equality conditions in the inequalities for regular and semiregular graphs.
• Reevaluates the traditional notion of graph irregularity by comparing the performance of Zagreb-based measures against known structural properties.

Experimental results

Research questions

• RQ1 Under what conditions do the inequalities involving M₁(G), M₂(G), and F(G) achieve equality in connected graphs?
• RQ2 How can the concept of semiregular graphs be generalized to include weakly semiregular graphs with at most two distinct vertex degrees?
• RQ3 To what extent do Zagreb index-based irregularity measures predict the stability of C40 fullerene isomers when applied to their dual graphs?
• RQ4 How do the proposed inequalities and index-based measures compare to traditional irregularity characterization methods in graph theory?
• RQ5 What insights does the analysis of dual graphs of C40 fullerene isomers provide for refining topological descriptors in chemical graph theory?

Key findings

• Equality in the derived inequalities between M₁(G), M₂(G), and F(G) holds if and only if the graph G is regular or semiregular.
• The proposed notion of weakly semiregular graphs extends the class of bidegreed semiregular graphs by relaxing the requirement of uniform degree distribution across partitions.
• Comparative tests on dual graphs of C40 fullerene isomers show that Zagreb index-based irregularity indices can effectively distinguish structural features relevant to stability.
• The study reveals that traditional graph irregularity characterization may be insufficient, and that Zagreb index-based measures offer a more nuanced and predictive alternative.
• The results support the use of M₁, M₂, and F indices as reliable topological descriptors in the analysis of molecular graphs, especially in nanomaterials like fullerenes.
• The analysis confirms that the inequality framework is robust and provides a theoretical foundation for further applications in chemical and materials science.