[Paper Review] On comparing Zagreb indices

Aleksandar Ilić, Dragan Stevanović|arXiv (Cornell University)|Apr 21, 2011

Graph theory and applications66 citations

TL;DR

This paper investigates the conjecture that M1/n ≤ M2/m for all simple connected graphs, proving it holds for regular graphs and subdivision graphs, but demonstrating it fails for connected graphs with ≥2 cycles. The authors establish sharp common lower and upper bounds for M1/n and M2/m, showing equality only in regular graphs, and construct explicit counterexamples with arbitrary cycle counts, proving the conjecture requires structural constraints like bounded cycle count or maximum degree.

ABSTRACT

Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j) \in E} d_i d_j$, where $d_i$ denote the degree of vertex $i$. Recently proposed conjecture $M_1 / n \leqslant M_2 / m$ has been proven to hold for trees, unicyclic graphs and chemical graphs, while counterexamples were found for both connected and disconnected graphs. Our goal is twofold, both in favor of a conjecture and against it. Firstly, we show that the expressions $M_1/n$ and $M_2/m$ have the same lower and upper bounds, which attain equality for and only for regular graphs. We also establish sharp lower bound for variable first and second Zagreb indices. Secondly, we show that for any fixed number $k\geqslant 2$, there exists a connected graph with $k$ cycles for which $M_1/n>M_2/m$ holds, effectively showing that the conjecture cannot hold unless there exists some kind of limitation on the number of cycles or the maximum vertex degree in a graph. In particular, we show that the conjecture holds for subdivision graphs.

Motivation & Objective

• To resolve the conjecture M1/n ≤ M2/m for general simple graphs.
• To identify structural conditions under which the conjecture holds or fails.
• To establish sharp lower and upper bounds for M1/n and M2/m, showing equality holds only for regular graphs.
• To investigate the behavior of variable Zagreb indices and derive their bounds.
• To construct counterexamples with arbitrary numbers of cycles to show the conjecture cannot hold universally without constraints.

Proposed method

• Apply the Cauchy-Schwarz inequality to derive the sharp lower bound M1 ≥ 4m²/n, with equality iff G is regular.
• Use the AM-GM and Jensen’s inequality on logarithmic functions to prove M2 ≥ 4m³/n², with equality iff G is regular.
• Derive upper bounds for M1/n and M2/m in terms of maximum degree Δ and minimum degree δ, showing equality iff G is regular.
• Construct a family of graphs C(a,b) with a star and b triangles attached in a line to generate counterexamples with b cycles.
• Analyze the quadratic inequality in a for M2·n − M1·m ≥ 0 to determine when the conjecture fails.
• Prove the conjecture holds for subdivision graphs by showing M1(S(G))/(n+m) ≤ M2(S(G))/(2m), with equality iff G is regular.

Experimental results

Research questions

• RQ1 Does the inequality M1/n ≤ M2/m hold for all connected simple graphs?
• RQ2 What are the sharp lower and upper bounds for M1/n and M2/m, and when are they achieved?
• RQ3 Can counterexamples to the conjecture be constructed with an arbitrary number of cycles?
• RQ4 Under what graph structures does the conjecture M1/n ≤ M2/m hold?
• RQ5 Does the conjecture hold for subdivision graphs of arbitrary graphs?

Key findings

• The expressions M1/n and M2/m share the same sharp lower bound 4m²/n², achieved if and only if the graph is regular.
• The expressions M1/n and M2/m share the same sharp upper bound ΔM1/(2m), achieved if and only if the graph is regular.
• For any fixed b ≥ 2, there exists a connected graph with b cycles such that M1/n > M2/m, disproving the conjecture without structural constraints.
• The conjecture M1/n ≤ M2/m holds for subdivision graphs, with equality if and only if the original graph is regular.
• The counterexample C(12,2) with 19 vertices and 58 edges is the smallest known connected counterexample with 2 cycles.
• The variable Zagreb indices λM1 and λM2 satisfy λM1 ≥ n(2m/n)²λ and λM2 ≥ m(2m/n)²λ, with equality iff the graph is regular.