threshold graph

threshold graph
Name	Threshold graph
Edges	0..n(n-1)/2
Named after	C. T. (#?)
Introduced	1973
Related	Split graph, Cograph, Chordal graph, Interval graph

threshold graph

A threshold graph is a graph class defined by a simple additive rule producing graphs that are simultaneously sparse and dense extremes; it has tight connections to combinatorics, optimization, algebra, and theoretical computer science. These graphs admit multiple equivalent descriptions linking structural graph theory, linear inequalities, and forbidden induced subgraphs, and they appear in studies by researchers associated with Princeton University, Bell Labs, MIT, and other institutions. They provide canonical examples in the study of graph algorithms, spectral theory, and extremal combinatorics.

Definition and basic properties

A threshold graph can be defined via a weight function and a threshold inequality inspired by models from Economics such as Samuelson Prize-adjacent prize models and by probabilistic models used at Bell Labs; it is closed under induced subgraphs and complementation and is a perfect graph related to the Strong Perfect Graph Theorem. Basic properties include hereditary closure, degree sequence characterizations tied to Erdős–Gallai theorem-style constraints, and simple eigenvalue multiplicity patterns studied in spectral graph theory at institutions like Stanford University and University of Cambridge.

Equivalent characterizations

Multiple equivalent characterizations tie threshold graphs to forbidden configurations, construction sequences, and matrix inequalities that link to linear programming research at IBM Research and optimization work by scholars connected to INFORMS. Equivalences include forbidden induced subgraphs isomorphic to small graphs studied in classic papers from Princeton University and descriptions via 0-1 sequences related to work by researchers affiliated with Harvard University and University of California, Berkeley. Algebraic characterizations relate to adjacency matrix rank and to canonical forms used in papers from Columbia University.

Construction and generation methods

Construction methods permit generation by iterative addition rules analogous to processes described in combinatorial algorithm literature from MIT and Carnegie Mellon University. One method uses binary sequences that correspond to operations named in algorithmic graph theory seminars at ETH Zurich and École Polytechnique, while other generation schemes exploit degree sequence realizations connected to classical results discussed at University of Oxford and University of Tokyo.

Graph parameters and invariants

Parameters such as chromatic number, clique number, independence number, and matching number admit closed-form expressions for threshold graphs, matching combinatorial optimization topics from INFORMS conferences and program committees at SIAM. Spectral invariants show constrained eigenvalue spectra explored by groups at Princeton University and Caltech, and graph energy computations have been subjects of articles in journals edited by scholars from Elsevier and Springer.

Algorithmic problems and complexity

Many algorithmic problems are solvable in linear time on threshold graphs, making them benchmarks in algorithm design courses at MIT, Stanford University, and Harvard University. Problems such as maximum clique, vertex cover, and Hamiltonian path reduce to simple greedy rules; complexity contrasts with NP-complete problems studied historically in complexity theory at Bell Labs and University of California, Berkeley.

Applications and examples

Threshold graphs model simple resource allocation and binary decision systems similar to economic threshold models taught in curricula at London School of Economics and Yale University. Example graphs appear in network modeling case studies from Google Research and in theoretical examples in texts published by academic presses including Cambridge University Press and Oxford University Press.

History and related graph classes

The concept emerged in the early 1970s and has been developed by researchers affiliated with institutions such as Princeton University, Bell Labs, and University of Waterloo, with connections to related classes like Split graph, Cograph, Chordal graph, and Interval graph discussed in monographs and conference proceedings organized by ACM and SIAM. Subsequent work linked threshold graphs to studies in extremal graph theory appearing at venues including the European Conference on Combinatorics, the International Congress of Mathematicians, and workshops at Institut des Hautes Études Scientifiques.

Category:Graph theory