Albertson Index of Trees

• The Albertson index of trees is a topological invariant that quantifies the degree imbalance by summing the absolute differences between the degrees at the endpoints of each edge.
• It features explicit closed-form formulas for both general and caterpillar trees, with path trees minimizing and star trees maximizing the index.
• Its close ties to classical invariants like the Zagreb index make it a benchmark for assessing extremal degree-sequence properties in algebraic graph theory.

The Albertson index of a tree is a topological irregularity invariant introduced to quantitatively capture the extent of degree imbalance across a tree’s edges. Denoted $\operatorname{irr}(T)$ for a tree $T$, it is defined as the sum of absolute differences of degrees at endpoints of every edge. This index plays a central role in extremal and degree-sequence problems in algebraic graph theory, quantifying how far a tree is from regularity and serving as a benchmark for more complex irregularity indices. Its precise analytic form, sharp extremal bounds, and relationships to classical polynomial graph invariants have been the subject of recent research, with particular focus on explicit degree-sequence formulas and the interplay between structural arrangements and irregularity (Hamoud et al., 18 May 2025, Hamoud et al., 4 Dec 2025).

1. Formal Definition and Interpretations

For a simple connected tree %%%%2%%%%, the Albertson index is given by

$$\operatorname{irr}(T) = \sum_{uv\in E(T)} \left|\deg_T(u) - \deg_T(v)\right|$$

where $\deg_T(u)$ denotes the degree of vertex $u$. This sum reflects the "local degree disparity" at each edge and vanishes if and only if $T$ is regular (not possible for $n>2$ in trees).

Alternative interpretations arise in the context of degree sequence analysis and majorization: $\operatorname{irr}(T)$ is the minimal $L^1$ norm over all edge degree-difference assignments consistent with the given degree sequence and incidence structure (Hamoud et al., 22 Oct 2025).

2. Explicit Formulas: Degree Sequences and Caterpillars

If $(d_1,\dots,d_n)$ is the degree sequence of $T$, several closed forms are available for $\operatorname{irr}(T)$, most notably for caterpillar trees and generic trees:

• General Trees (sorted non-increasing degree order):

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

or, when all internal degrees coincide at $\lambda = (\Delta+\delta)/2$ (with $\Delta = d_1$ and $\delta = d_n$),

$$\operatorname{irr}(T) = d_1^2 + d_n^2 + (n-2)\left(\frac{\Delta+\delta}{2}\right)^2 + \sum_{i=2}^{n-1} d_i + d_n - d_1 - 2n - 2$$

• Caterpillar Trees:

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

This formula isolates backbone–leaf and backbone–backbone contributions, naturally extending to characterize extremal arrangements for given sequences (Hamoud et al., 4 Dec 2025, Hamoud et al., 22 Oct 2025).

3. Extremal Values: Path and Star Trees

The extremal trees with respect to the Albertson index are:

• Path $P_n$: Minimizes $\operatorname{irr}(T)$ for all trees of order $n$.
  • $\operatorname{irr}(P_n) = 2$.
• Star $S_n = K_{1,n-1}$: Maximizes $\operatorname{irr}(T)$ for all trees of order $n$.
  • $\operatorname{irr}(S_n) = (n-1)(n-2)$.

Any deviation from these structures—by distributing leaves more evenly or clustering them—interpolates between these extremal values. In particular, among all trees with a fixed degree sequence, the minimizer is the unique monotone caterpillar and the maximizer is a caterpillar with alternating large and small degree placements along the backbone (Hamoud et al., 4 Dec 2025, Hamoud et al., 18 May 2025, Hamoud et al., 22 Oct 2025).

4. Sharp Analytic Bounds and Equality Cases

Numerous bounds have been established, some tight only for special classes:

• General lower bound (in terms of Zagreb index):

$$\operatorname{irr}(T) \geq M_1(T) + \Delta(\Delta-1)$$

where $M_1(T) = \sum_v d_v^2$ is the first Zagreb index.

• Degree-sequence-dependent quadratic/cubic bounds (Hamoud et al., 4 Dec 2025):

$$\operatorname{irr}(T) \geq \sum_{i=1}^n d_i^2 + 2d_1 + d_2 - 3d_3 + 2(d_{n-1} + d_n) + \frac{\Delta}{4}$$

$$\operatorname{irr}(T) \geq \frac{1}{\alpha} \left( \sum_{i=1}^n d_i^3 - \frac{\beta m(m+1)}{\alpha} \right)$$

with $\alpha = \frac{d_1+d_2}{2}$, $\beta = \frac{d_{n-1}+d_n}{2}$, $m=n-1$.

• Extremal degree gap bounds (Hamoud et al., 9 Jun 2025):

$2 \leq \operatorname{irr}(T) \leq (n-1)(n-2)$

with equality for the path and star, respectively.

Table: Key Explicit Bounds

Bound Type	Formula	Equality Attainable For
Basic upper/lower	$2\leq \operatorname{irr}(T)\leq(n-1)(n-2)$	Path (min) / Star (max)
Zagreb-index lower	$\operatorname{irr}(T)\geq M_1(T)+\Delta(\Delta-1)$	Star
Quadratic degree lower	As above (see full formula)	Monotone caterpillar
Minimum given max deg	$$\operatorname{irr}_{\min} \geq \Delta^2(\Delta-1)2^{\alpha}/(\Delta-p)^2$$	(Hamoud et al., 19 Jul 2025)

5. Connections to Classical Indices and Degree Arrangements

A major structural insight is the direct formula linking the Albertson index to the first Zagreb index and degree sums: $$\operatorname{irr}(T) = M_1(T) + \sum_{i=2}^{n-1} d_i + d_n - d_1 - 2n - 2$$ This connection enables the transfer of analytic results (for example, extremal Zagreb index problems) to the context of edge-degree-imbalance. In addition, the relationship to the Sigma index (quadratic in degree-differences) admits comparative inequalities such as $$\sqrt{\sigma(T)} \leq \operatorname{irr}(T) \leq \sqrt{m\sigma(T)}$$ (Hamoud et al., 13 Feb 2025), and for stars, $\sigma(S_n) = (n-2)\operatorname{irr}(S_n)$ (Hamoud et al., 9 Jun 2025). The explicit dependence on the degree arrangement is exploited in rearrangement inequalities and majorization-type proofs, determining exactly which tree realization of a given sequence is extremal (Hamoud et al., 22 Oct 2025).

6. Structural Dependence and Computation

Given any degree sequence $(d_1,\ldots,d_n)$, computation of $\operatorname{irr}(T)$ uses the explicit closed form, requiring $O(n)$ operations. For large trees, the midpoint approximation (using average internal degree) is highly accurate: $$\operatorname{irr}(T)\approx d_1^2+d_n^2+(n-2)\left(\frac{\Delta+\delta}{2}\right)^2+\frac{2(n-1)-d_1-d_n}{2} + d_n - d_1 - 2n -2$$ Even in the absence of the precise internal arrangement, sharp bounds are achievable in terms of $n$, $\Delta$, $\delta$, and classical invariants, ensuring that the degree sequence alone provides tight control on irregularity (Hamoud et al., 18 May 2025, Hamoud et al., 4 Dec 2025).

7. Significance, Applications, and Extensions

The Albertson index serves as a fundamental graph irregularity measure, underpinning broader families of topological indices and offering a concrete characterization of irregularity extremals in combinatorics and mathematical chemistry. Its analytic tractability and sensitivity to degree arrangements have made it standard for benchmarking irregularity, for proving degree-sequence extremal results, and as a prototype for $p$-norm or higher-order irregularity indices (such as the Sigma index or the general $A_p$ index (Lin et al., 2021)). Research continues into its extremal configurations under various constraints, refined bounds for subclasses of trees (e.g., caterpillars, Bethe trees, Fibonacci degree trees), and interplay with other graph invariants (Hamoud et al., 4 Dec 2025, Hamoud et al., 22 Oct 2025, Hamoud et al., 12 Jun 2025, Hamoud et al., 1 May 2025).