Albertson Index: Graph Irregularity Measure

• Albertson Index is a degree-based invariant that sums the absolute differences between the degrees of adjacent vertices to measure graph irregularity.
• It is zero for regular graphs and maximal for structures like star and caterpillar trees, providing clear bounds on network heterogeneity.
• Its analytical framework leverages degree sequences for efficient computation, playing a vital role in chemical graph theory and network analysis.

The Albertson Index quantifies the degree irregularity of a graph by summing the absolute differences between the degrees of the endpoints of each edge. Formally, for a simple undirected graph $G = (V, E)$ with vertex degrees $d(u)$, the Albertson Index is defined as

$\mathrm{irr}(G) = \sum_{uv\in E} |d(u) - d(v)|.$

This measure vanishes for regular graphs and increases with the disparity in node degrees, making it a central tool in the analysis of irregularity for chemical graph theory, extremal graph theory, and network structure characterization.

1. Mathematical Formulation and Basic Properties

The Albertson Index is a degree-based edge sum invariant:

• For each edge $uv$, the contribution is $|d(u) - d(v)|$.
• $\mathrm{irr}(G) = 0$ iff $G$ is regular.
• For trees, particularly star graphs $S_n$ (one vertex of degree $n-1$ and $n-1$ leaves), the index reaches its maximal value: $\mathrm{irr}(S_n) = (n-2)(n-1)$ (Hamoud et al., 19 Jul 2025).
• For caterpillar trees, explicit closed-form formulas exist, e.g., $\mathrm{irr}(\mathscr{C}(n,m)) = m(m+1)n - 2m + 2$ for $n\geq 3$ (Hamoud et al., 22 Oct 2025).

For arbitrary degree sequences $\mathscr{D} = (d_1, \dots, d_n)$, the index can be written as

$$\mathrm{irr}(T) = \sum_{i=1}^{n} (d_i - 1)(d_i - 2) + d_1 + d_n + 2 + \sum_{i=1}^{n-1} |d_i - d_{i+1}|$$

for an ordered tree degree sequence, which isolates quadratic and linear contributions to irregularity (Hamoud et al., 22 Oct 2025).

2. Extremal Behavior and Bounds

The Albertson Index varies sharply with graph structure:

• Maximum Irregularity: Achieved by star graphs; for trees, $\mathrm{irr}(S_n) = (n-2)(n-1)$ (Hamoud et al., 9 Jun 2025).
• Minimum Irregularity: Attained by path graphs $P_n$, which have all internal vertices of degree 2 (Hamoud et al., 19 Jul 2025).
• For caterpillar trees, specific formulas and extremal configurations are determined by the arrangement of the degree sequence; e.g., for $m=3$, $\mathrm{irr}(C(n,3)) = 12n-4$ (Hamoud et al., 13 Feb 2025).

Sharp bounds include: $$\mathrm{irr}_{\min} \geq \Delta^2(\Delta-1) \frac{2^\alpha}{(\Delta-p)^2}$$ for trees with given maximum degree $\Delta$, integers $p, \alpha$ satisfying $1\leq p\leq \alpha\leq \Delta-3$ (Hamoud et al., 19 Jul 2025). Additionally,

$$\mathrm{irr}(G) > \frac{\delta (\Delta-\delta)^2 |V|}{\Delta + 1}$$

where $\delta$, $\Delta$ are minimum and maximum degrees, placing a lower bound given the degree extremes (Hamoud et al., 9 Jun 2025).

In bipartite graphs (with partition sizes $|V_1| = n_1, |V_2| = n_2$), explicit extremal formulas and parametric bounds facilitated by variables such as

$$\lambda = \frac{4}{3} n_1 - \sqrt{\frac{28}{9} n_1^2 - \frac{8}{3} n_1 n_2}$$

yield precise estimates for $\mathrm{irr}_{\min}, \mathrm{irr}_{\max}$ (Hamoud et al., 19 Jul 2025).

3. Analytical Frameworks and Structural Implications

Analytical derivations decompose the Albertson Index into quadratic and linear degree sequence sums (Hamoud et al., 22 Oct 2025). The methodology capitalizes on ordering the degree sequence and summing local and global contributions, for example,

• Quadratic sum: $(d_i-1)(d_i-2)$, sensitive to vertices with high degrees.
• Linear sum: $|d_i-d_{i+1}|$, records jumps in degree ordering.

This framework explains the rigidity in irregularity for some classes, e.g., specific caterpillars, where maximum and minimum values coincide (Hamoud et al., 13 Feb 2025).

Approximate computation becomes tractable by recasting the problem in terms of degree sequences rather than enumerating all non-isomorphic graphs (Abdo et al., 2012). This procedure is especially useful for large graphs or in applications requiring rapid estimation.

4. Connections to Other Indices

• Sigma Index: The quadratic analog, $\sigma(G) = \sum_{uv\in E} (d(u)-d(v))^2$, amplifies the effect of higher degree differences and has lower bounds closely tied to $\mathrm{irr}(G)$: $$\sigma(T) \geq \mathrm{irr}(T) + \left\lfloor\frac{n-2}{a_r-t_m}\right\rfloor + \Delta(\Delta_{\mathscr{A}} - \Delta_{\mathscr{R}})^2$$ (Hamoud et al., 23 Sep 2025).
• Zagreb Indices: The Albertson Index bounds (both upper and lower) relate to first and second Zagreb indices, relevant in chemical graph theory (Hamoud et al., 22 Oct 2025, Avdullahu et al., 2022).
• Sombor and Related Indices: Geometric–arithmetic generalizations (e.g., SO₁, diminished Sombor index DSO) embed the Albertson term as a factor and allow for bounds such as

$$\frac{\sqrt{2}}{4\Delta}\cdot\mathrm{irr}(G) + \frac12\mathrm{GA}(G) \leq \mathrm{DSO}(G) \leq \frac{1}{2\delta}\cdot\mathrm{irr}(G) + \frac{\sqrt{2}}{2}\mathrm{GA}(G)$$

with extremal cases precisely characterized for regular or edgeless graphs (Movahedi, 3 Aug 2025, Ghanbari et al., 2022).

5. Applications in Chemical Graph Theory and Network Analysis

The Albertson Index (also referenced as the third Zagreb index in chemical applications) is critical for:

• Bounding classic molecular descriptors (such as the first and second Zagreb indices), linking molecular topology to chemical reactivity and stability (Abdo et al., 2012, Avdullahu et al., 2022).
• Characterizing the branching patterns and backbone irregularity in molecules modeled as trees or caterpillars (Hamoud et al., 22 Oct 2025, Hamoud et al., 13 Feb 2025).
• Assessing vulnerability and heterogeneity in networks, where high irregularity may signal non-uniform load distribution or critical connectivity structures (Hamoud et al., 9 Jun 2025).

Efficient computational methods leveraging degree sequences and closed-form formulas enable practical evaluation in large-scale graphs as needed for molecular modeling and network science.

6. Recent Generalizations and Extremal Graph Constructions

Recent works develop generalized irregularity indices such as the $p$-norm Albertson index,

$$A_p(G) = \left(\sum_{uv\in E}|d(u)-d(v)|^p\right)^{1/p}$$

which interpolates between the classical (linear) and quadratic (sigma) indices, providing extremal bounds and explicit formulas for structured trees (such as Bethe and Kragujevac trees) (Lin et al., 2021).

Maximally irregular configurations (e.g., clique–star graphs) achieve sharp upper bounds, and transformation techniques (e.g., rearranging vertices, promoting universality) systematically produce graphs maximizing irregularity in various cyclic or degree-constrained families (Abdo et al., 2012).

7. Impact on Extremal and Algorithmic Graph Theory

The Albertson Index serves as a key comparative tool in extremal graph theory, guiding the identification of graphs with maximal or minimal irregularity for given parameters. Algorithmic implications include:

• Practical irregularity estimation for large graphs via degree sequence-based methodologies.
• Direct computation for fixed degree sequences through unified analytical formulas.
• Quantitative discrimination between competitive chemical or network structures with similar degree distributions but distinct connectivity properties.

These advances foster deeper exploration of irregularity as a structural and functional property in both abstract and applied graph-theoretic research.

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Key research references: (Abdo et al., 2012, Hamoud et al., 9 Jun 2025, Hamoud et al., 22 Oct 2025, Hoster et al., 1 Oct 2025, Hamoud et al., 19 Jul 2025, Avdullahu et al., 2022, Hamoud et al., 23 Sep 2025, Lin et al., 2021, Ghanbari et al., 2022, Movahedi, 3 Aug 2025, Hamoud et al., 13 Feb 2025, Hamoud et al., 12 May 2025, Hoster et al., 1 Oct 2025).