Threshold Graphs Overview

• Threshold graphs are finite, simple graphs defined via vertex weight inequalities, forbidden induced subgraphs, or binary construction sequences.
• They exhibit strong algebraic and combinatorial structures, characterized by totally ordered neighborhood systems and unique degree sequences.
• Their robust spectral properties and efficient O(n) algorithms underpin significant applications in network dynamics, optimization, and multi-agent systems.

A threshold graph is a finite, simple graph admitting equivalent definitions by forbidden induced subgraphs, vertex-weight inequalities, or binary construction sequences. Threshold graphs are uniquely determined by a recursive construction—at each step, a new vertex is added as either isolated or dominating with respect to the current vertex set, leading to strong algebraic and combinatorial structure. This class is characterized as the family of graphs with no induced $P_4$, $C_4$, or $2K_2$, equivalently as graphs whose neighborhood system is totally (vicinal) ordered by inclusion, and as split graphs with a nested bipartite part. The threshold graph concept has spawned a rich theory of combinatorial structure, spectral characterization, extremal invariants, and algorithmic applications.

1. Definitions and Equivalent Characterizations

Threshold graphs $G=(V,E)$ admit the following equivalent descriptions:

• Vertex weight model: There exists $w:V\to\mathbb{R}$ and threshold $t\in\mathbb{R}$ such that $uv\in E$ if and only if $w(u)+w(v)\ge t$.
• Forbidden subgraph characterization: $G$ is threshold if and only if it contains no induced subgraph isomorphic to $P_4$ (4-vertex path), $C_4$ (4-cycle), or $2K_2$ (two disjoint edges) (Helmberg et al., 2023, Raja et al., 2022, Ravanmehr et al., 2016).
• Binary (0/1) construction: The vertex sequence is uniquely encoded by $b=\alpha_1\ldots\alpha_n$ with $\alpha_1=0$ and each $\alpha_i\in\{0,1\}$, where $\alpha_i=1$ means addition as a dominating vertex, $\alpha_i=0$ as isolated. The final vertex must be dominating for connectedness (Raja et al., 2022).
• Split and nested structure: $G$ is split into an independent set $U$ and a clique $V$, with a unique partition $U=U_1\sqcup\dots\sqcup U_h$, $V=V_1\sqcup\dots\sqcup V_h$, and edges from $U_i$ to $V_j$ precisely if $j\ge i$ (Anđelić et al., 2018).

All induced subgraphs of a threshold graph are themselves threshold. The family is exactly the intersection of split graphs, cographs (P$_4$-free), and interval graphs (Raja et al., 2022). Threshold graphs arise as extremal graphs in combinatorial optimization and network modeling (Ravanmehr et al., 2016).

2. Enumerative and Structural Properties

The number of labeled threshold graphs on $n$ vertices is $2^{n-1}$; each corresponds to a binary string of length $n$ with prescribed initial/final conditions. For structural invariants:

• Degree sequences: All vertices in the same block of consecutive 0s or 1s in the construction have equal degree. The degree sequence (sorted nonincreasingly) is often strictly staircase-shaped.
• Neighborhood ordering: The family $\{N[v]:v\in V\}$ is totally ordered by inclusion; this property induces a Ferrers diagram majorization (Helmberg et al., 2023).
• Metric dimension: For block-encoded string $0^{s_1}1^{t_1}\ldots 0^{s_k}1^{t_k}$, $B(G)=\sum_{i=1}^k (s_i+t_i-2)$. An $O(n)$ time algorithm achieves this using only the binary sequence (Raja et al., 2022).

Enumeration can be refined by number of dominating vertices, number of components, and other statistics. The number of connected threshold graphs is exactly half that of all threshold graphs (Galvin et al., 2021).

3. Spectral Theory

Adjacency and Related Matrices

• Adjacency matrix $A$: The characteristic polynomial can be computed recursively from the binary sequence, with explicit formulas for the multiplicities of eigenvalues 0 and $-1$ (Lazzarin et al., 2018). Every non-isomorphic threshold graph has a unique spectrum (i.e., no cospectral pairs).
• Seidel matrix $S = J - I - 2A$: The characteristic polynomial of $S$ admits a recurrence based on the $b$-sequence, and allows explicit calculation of determinant and of blockwise multiplicities for $\pm1$ as Seidel eigenvalues. Threshold graphs with at most five distinct Seidel eigenvalues are classified explicitly. Families of nonisomorphic, Seidel-cospectral threshold graphs exist (Mandal et al., 2021).
• Normalized adjacency matrix $\mathcal A$: The normalized spectrum splits into two parts—eigenvalues arising directly from the consecutive runs in $b$, and the simple spectrum of a normalized partition matrix. The total number of distinct normalized eigenvalues is bounded in terms of the block structure; energy bounds depend on block sizes. Complete and star graphs attain the minima/optima for number of distinct eigenvalues (Banerjee et al., 2016).
• Signless Laplacian $Q=D+A$: In threshold graphs, $Q$ has a large subspace of "trivial" eigenvectors from block-difference vectors; the remaining spectrum aligns with that of a condensed matrix determined by the partition. The spectrum of $Q$ interlaces exactly with the degree sequence, providing the optimal confirmation of the signless Brouwer conjecture for this class (Helmberg et al., 2023).
• Laplacian $L=D-A$: The Laplacian spectrum is determined explicitly by the conjugate degree sequence. Fast $O(n)$ algorithms yield full eigenbases, with all eigenvalues integer; these encode structural features such as symmetry and block multiplicity. Control-theoretic considerations (see below) follow from this spectral structure (Hsu, 2017).

Vertex-Type Theory

In the adjacency spectrum, the so-called downer, neutral, and Parter vertex types can be completely classified for threshold graphs. For nontrivial simple eigenvalues, almost all vertices are downers. Neutral vertices, if present, are highly constrained to special blocks, determined by the nesting pattern of the split partition (Anđelić et al., 2018).

4. Generalizations, Multivariate Extensions, and Related Classes

Paired Threshold (PT) Graphs

PT graphs interpolate between threshold and unit interval graphs by combining a sum threshold and a difference threshold. A PT graph with parameters $(T_\Sigma, T_\Delta)$ admits a vertex weighting so that $uv\in E$ precisely when $w(u)+w(v)\ge T_\Sigma$ and $|w(u)-w(v)|\le T_\Delta$. PT graphs are chordal, admit a layered decomposition involving threshold components, and comply with new homophily constraints. Recognizing PT graphs is polynomial-time via 2-SAT encoding. Threshold graphs are precisely the PT graphs where $T_\Delta\to\infty$ (Ravanmehr et al., 2016).

$k$-Threshold Graphs

A $k$-threshold graph is defined by the parity of the number of thresholds exceeded by each edge, generalizing classical threshold graphs (the $k=1$ case) (Wang, 17 Jun 2024). The threshold number $\operatorname{tn}(G)$ is the minimal $k$ such that $G$ can be represented as a $k$-threshold graph. Explicit formulas and sharp bounds for $\operatorname{tn}(G)$ on path-related graphs, multipartite, and cluster graphs are known (Wang, 17 Jun 2024).

H-Product and H-Threshold Graphs

The class of H-threshold graphs further generalizes threshold graphs to structures grown from a binary operation ($\circ_H$) dictated by a digraph $H$ on $k$ vertices. An H-threshold graph is a repeated H-product of one-vertex $k$-partitioned graphs. The threshold-width $\mathrm{ThrWidth}(G)$ quantifies the minimal $k$ needed for this construction. Threshold and difference graphs correspond exactly to H-threshold graphs for $k=2$, with specific digraph choices. The theory extends unique factorization principles and provides characterization for $\mathrm{ThrWidth}=1$ or $2$ (Skums, 2010).

Oriented Threshold Graphs

Oriented threshold graphs generalize to directed graphs using ternary (0,$+$,$-$) construction, weight inequalities with directionality, and nested in/out-neighborhoods. The number of oriented threshold graphs on $n$ vertices is the $2n$th Fibonacci number. Underlying equivalences extend to the directed setting and characterize these graphs as transitive orientations of undirected threshold graphs (Boeckner, 2015).

5. Algorithms and Invariant Computation

Key algorithmic results include:

• Metric dimension: Linear-time calculation from the run-length code (Raja et al., 2022).
• Laplacian eigenvector basis: $O(n)$ algorithm provided for threshold graphs based on block structure and conjugate degree sequence (Hsu, 2017).
• Recognition: Classical threshold graphs are recognized in $O(n)$ using forbidden subgraph or binary sequence construction; PT graphs can be recognized via explicit 2-SAT reductions and structural decomposition (Ravanmehr et al., 2016).
• Component and dominating-vertex enumeration: Closed, bijectively justified formulas enumerate labeled threshold graphs by number of components, dominating vertices, etc. (Galvin et al., 2021).
• Exact coloring/invariants: The $L(2,1)$-chromatic number and restricted threshold dimension admit closed recurrent expressions via the block encoding (Raja et al., 2022).
• Controllability: For leader-follower Laplacian systems, controllability reduces to the maximum multiplicity of Laplacian eigenvalues. Class-specific binary control matrices that meet this bound can be easily constructed. Explicit construction methods for graphs achieving minimal controllability constraints (e.g., combining two antiregular graphs) are described (Hsu, 2017).

6. Dynamics, Synchronization, and Physical Applications

Threshold graphs possess remarkable dynamical properties:

• Kuramoto synchronization: Every connected threshold graph is globally synchronizing for the homogeneous Kuramoto model (identical oscillators), i.e., irrespective of initial conditions, the coupled oscillator system converges to full synchrony. The proof relies on the existence and stability of geometric symmetries in the plane driven by the nested neighborhood structure. This global synchronization is not attributable to density but to combinatorial regularity; it is witnessed for both highly sparse (stars) and dense (cliques) threshold graphs (Wu et al., 16 Nov 2025).
• Consensus and multi-agent systems: Threshold graphs support optimal controllability and minimal input requirements, as shown in Laplacian-based consensus protocols. Their block structure is advantageous for the design of networks with specified dynamic behavior under edge or input constraints (Hsu, 2017).

7. Further Developments and Open Directions

Current research extends the threshold graph paradigm along multiple lines:

• Strong spectral interlacing: The signless Laplacian spectrum's tight interlacing with degree sequences is conjectured to admit generalization to other hereditary or degree-determined graph classes, such as split graphs and cographs (Helmberg et al., 2023).
• Partition-based majorization: Ferrers-majorization underlies fine-grained spectral and combinatorial properties, providing a bridge between threshold graphs and broader majorization inequalities in combinatorics.
• Multithreshold and H-product frameworks: Threshold-width and related invariant-minimization problems connect threshold graphs to finite decompositions, recognition complexity, and forbidden structure theory (Skums, 2010, Wang, 17 Jun 2024).
• Enumerative refinements: The rich association to Eulerian and Stirling number triangles, ordered partitions, and the exponential formula provides a template for studying close relatives such as quasi-threshold, loop-threshold, and related graph classes (Galvin et al., 2021).
• Directed/weighted analogues: The orientation, weight-assignment, and transitive-extension perspectives suggest open questions for network structure learning, digraph limits, and extremal directed graph theory (Boeckner, 2015).

Threshold graphs remain a cornerstone in algebraic, combinatorial, and algorithmic graph theory, with new implications for dynamics on networks, invariant design, and expansion to generalized and multivariate models.