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| path graph | |
|---|---|
| Name | Path graph |
| Edges | n-1 |
| Degree | 1 or 2 |
| Chromatic number | 2 (for n>1) |
path graph
A path graph is a simple undirected graph formed by a finite sequence of distinct vertices connected by edges in a single line. It serves as a fundamental example in graph theory and combinatorics and appears in studies associated with Leonhard Euler, Arthur Cayley, Paul Erdős, Brendan McKay, and institutions such as the American Mathematical Society and the London Mathematical Society. The structure underpins results in fields influenced by Isaac Newton and Évariste Galois through algebraic graph theory and appears in applications linked to Alan Turing-era problems and modern work at Massachusetts Institute of Technology.
A path graph on n vertices is defined as a sequence v1, v2, ..., vn where each consecutive pair vi, vi+1 is adjacent and no other adjacencies occur. Formal treatments appear in texts from the European Mathematical Society and research by William Tutte and Claude Berge. Standard nomenclature contrasts the path graph with the Dijkstra algorithm-motivated shortest-path constructs and with cycle graphs studied by Arthur Cayley.
Vertices at the sequence ends have degree one while internal vertices have degree two; these degree patterns are discussed in expositions by Paul Erdős and in course notes at University of Cambridge. The number of edges equals n−1 and the graph is acyclic and connected, making it a tree as treated in lectures by G. H. Hardy and in the work of Otakar Borůvka. Bipartiteness, planarity, and outerplanarity are immediate and are exploited in proofs associated with Kurt Gödel-inspired combinatorial constructions and projects at Princeton University.
Adjacency and Laplacian matrices of the path graph have tridiagonal forms studied in linear algebra courses at Harvard University and analyzed in papers by Israel Gelfand and Eugene Wigner. Eigenvalues are known explicitly in closed form, with connections to trigonometric functions and to classical orthogonal polynomials investigated by S. Ramanujan and cited in monographs from the Institute for Advanced Study. The algebraic multiplicity of eigenvalues, characteristic polynomials, and relations to the matrix-tree theorem appear in combinatorial treatments by William Tutte and in spectral graph theory seminars at Stanford University.
Related constructs include path unions, induced subpaths, and concatenations that lead to caterpillar trees discussed in surveys by Béla Bollobás and in algorithmic contexts at Carnegie Mellon University. The path graph contrasts with cycle graphs, star graphs, and complete graphs as treated in comparative studies by Paul Erdős and Van H. Vu. Generalizations appear in grid graphs used at NASA and in Cartesian products studied in conference proceedings of the IEEE and the Association for Computing Machinery.
Path graphs model linear chain molecules in research from Linus Pauling and appear in polymer physics work at Bell Labs and in computational chemistry groups at California Institute of Technology. In computer science, they represent linked-list structures central to algorithm analysis in textbooks from Donald Knuth and in data structure courses at Massachusetts Institute of Technology. They appear in network routing examples in standards developed by IETF and in scheduling problems studied by Edsger Dijkstra and groups at IBM Research.
Constructive generation via vertex insertion and edge addition is described in enumerative combinatorics by Richard Stanley and enumeration of labelled vs unlabeled paths is treated in classic work by Arthur Cayley and in modern computational enumeration by Brendan McKay. Counting nonisomorphic labelled paths and embedding counts on surfaces are topics covered in proceedings of the International Congress of Mathematicians and in texts from the Cambridge University Press.