Non-Increasing Degree Sequences

• Non-increasing degree sequence is a vector of non-negative integers sorted in descending order that represents vertex degrees in graphs and simplifies structural analysis.
• Graphicality is verified using the Erdős–Gallai criterion and its continuous analogue, ensuring the sequence can realize a simple graph.
• Enumeration methods, limit objects, and algorithmic strategies link non-increasing degree sequences to applications in network science, extremal combinatorics, and statistical models.

A non-increasing degree sequence is a vector of non-negative integers representing the degrees of the vertices in a graph, arranged in order so that each entry is at least as large as all entries that follow. If $d = (d_1, d_2, \dots, d_n)$ with $d_1 \geq d_2 \geq \dots \geq d_n$, then $d$ is called a non-increasing degree sequence. This convention is central in graph theory, network science, and random graph models, as it simplifies analytical techniques, enables concise characterization of structural constraints, and facilitates combinatorial and statistical analysis.

1. Graphicality and the Erdős–Gallai Criterion

A sequence $d_1 \geq d_2 \geq \dots \geq d_n$ is graphical (i.e., it is the degree sequence of a simple graph) if and only if it has even sum and satisfies the inequalities

$$\sum_{i=1}^k d_i \leq k(k-1) + \sum_{i=k+1}^n \min\{d_i, k\} \qquad \text{for all } 1 \leq k \leq n.$$

This is the classical Erdős–Gallai criterion, which is restated and generalized to continuous functions in (Chatterjee et al., 2010). If the non-increasing sequence lies sufficiently away from extremal boundaries and is suitably scaled, it admits an even more general continuous analogue: $$\int_x^1 \min\{f(y), x\} \, dy + x^2 - \int_0^x f(y) \, dy > 0 \qquad \forall x \in (0,1].$$ Here, $f$ is a scaling limit (a left-continuous non-increasing function on $[0,1]$) representing the degree sequence as $n \to \infty$.

For many classes of graphs (including bipartite, directed, and $r$-multigraphs), these criteria reduce to verifying inequalities only at the "corners"—indices where the value drops in the non-increasing sequence—yielding computational and theoretical efficiency (Miller, 2012).

2. Enumeration and Structural Characterization

The enumeration of graphs with a prescribed non-increasing degree sequence is tractable by recurrence relations. For labeled graphs with degree sequence $d = (d_1 \geq \dots \geq d_n)$,

$$C((d_1, \dots, d_n)) = \sum_{S \in \binom{[n-1]}{d_n}} C\left( (d_1 - \chi_S(1), \dots, d_{n-1} - \chi_S(n-1))^\downarrow \right),$$

where $\chi_S(i)$ is the indicator for $i \in S$ and $(\cdot)^\downarrow$ is the reordered sequence, recursively reduced (Kaygun, 2021). For large $n$ and "near-regular" profiles, the number of (simple) graphs realizing such a degree sequence admits the asymptotic formula (Liebenau et al., 2017): $$g(\mathbf{d}) \sim \sqrt{2}\, \exp \left(\frac{1}{4} - \frac{\gamma_2^2}{4\mu^2(1-\mu)^2}\right) \left( \mu^{\mu}(1-\mu)^{1-\mu} \right)^{\binom{n}{2}} \prod_{i=1}^n \binom{n-1}{d_i},$$ with $\gamma_2$ the scaled degree variance and $\mu$ the mean scaled degree.

3. Limit Objects and the β-Model

For dense non-increasing degree sequences, there is a unique graph limit (graphon) in the Lovász–Szegedy sense (Chatterjee et al., 2010): $$W(x, y) = \frac{e^{g(x) + g(y)}}{1 + e^{g(x) + g(y)}}, \qquad f(x) = \int_0^1 W(x, y) \, dy,$$ where $g$ is determined from the scaled degree sequence. The β-model for random graphs uses non-increasing degree sequences as sufficient statistics; it admits unique and consistent maximum likelihood estimation via the system

$$d_i = \sum_{j \neq i} \frac{e^{\beta_i + \beta_j}}{1 + e^{\beta_i + \beta_j}},$$

with a fixed-point iteration algorithm converging geometrically in $\|\cdot\|_\infty$.

4. Threshold Graphs and Adjacency Rigidity

A fundamental property of non-increasing degree sequences is their ability to force adjacency relationships. For a given degree sequence $d = (d_1, \dots, d_n)$, the pair $\{i, j\}$ is forced to be adjacent across all realizations if incrementing both $d_i$ and $d_j$ yields a non-graphic sequence, and forced non-adjacency if decrementing both does so (Barrus, 2015). The set of such sequences is upward-closed under dominance (majorization) order. For threshold graphs, every pair is forced, and the envelope (intersection and union) graphs derived from the sequence are themselves threshold, characterized structurally by the vanishing of Erdős–Gallai differences for relevant indices.

5. Topological Indices for Trees

For trees with non-increasing degree sequences $\mathscr{D} = (d_1, ..., d_n)$, explicit relationships between the first Zagreb index $M_1(T)$ and the Albertson index $\operatorname{irr}(T)$ are given (Hamoud et al., 18 May 2025): $$\operatorname{irr}(T) = M_1(T) + \sum_{i=2}^{n-1} d_i + d_n - d_1 - 2n - 2,$$ where $M_1(T) = \sum_{v \in V(T)} d(v)^2$. This formula makes use of the ordered degree sequence to directly relate global connectivity and irregularity measures.

6. Computational Complexity and Algorithmic Considerations

While deciding graphicality from a non-increasing degree sequence is polynomial time tractable, extensions to higher-order constraints (e.g., the second-order degree sequence) or polytope optimization are NP-hard (Erdős et al., 2016, Bach et al., 2023). Membership testing for degree-sequence polytopes and realization for "typical" sequences are tractable via linear programming with single exponential runtime, but boundary cases and global optimization remain computationally intractable.

7. Applications and Theoretical Implications

Non-increasing degree sequences serve as fundamental invariants in random graph models, extremal combinatorics, graph structure characterizations (threshold, split, DSF sets (Barrus et al., 2013)), spectral theory (Liu et al., 2012, Shuchao et al., 2012), and algorithmic graph enumeration. The reduction of degree sequence criteria to verification at key indices ("corners") increases practical efficiency in recognizing graphical sequences and classes of graphs (Miller, 2012). In extremal combinatorics, degree sequence criteria provide sharper tiling thresholds (Hyde et al., 2018), and in network science non-increasing degree sequences underpin the analysis of degree distributions, connectivity, and robustness.

This body of theory demonstrates that the non-increasing arrangement of degrees is not just a matter of convenience, but a powerful structural convention that underlies major analytic, algorithmic, and statistical results in modern graph theory.