Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H(d,q) has vertex set Sd, the set of ordered d-tuples of elements of S, or sequences of length d from S. Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph H(d,q) is, equivalently, the Cartesian product of d complete graphs Kq.[1]
In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.[3] Unlike the Hamming graphs H(d,q), the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.
^ abBrouwer, Andries E.; Haemers, Willem H. (2012), "12.3.1 Hamming graphs" (PDF), Spectra of graphs, Universitext, New York: Springer, p. 178, doi:10.1007/978-1-4614-1939-6, ISBN 978-1-4614-1938-9, MR 2882891, retrieved 2022-08-08.
^Karami, Hamed (2022), "Edge distance-balanced of Hamming graphs", Journal of Discrete Mathematical Sciences and Cryptography, 25: 2667–2672, doi:10.1080/09720529.2021.1914363.
^Imrich, Wilfried; Klavžar, Sandi (2000), "Hamming graphs", Product graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, pp. 104–106, ISBN 978-0-471-37039-0, MR 1788124.
their Hamming distance is one. The Hamminggraph H(d,q) is, equivalently, the Cartesian product of d complete graphs Kq. In some cases, Hamminggraphs may...
include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), Hamminggraph concepts...
graphs K2. More generally the Cartesian product of copies of a complete graph is called a Hamminggraph; the hypercube graphs are examples of Hamming...
3. For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions...
solid. An extension is the three dimensional k-ARY Hamminggraph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing...
graph. In bioinformatics, De Bruijn graphs are used for de novo assembly of sequencing reads into a genome. De Bruijn torus Hamminggraph Kautz graph...
distance, Hamming distance and Jaro–Winkler distance may be interpreted as graph edit distances between suitably constrained graphs. Likewise, graph edit distance...
hypercube graphs, and the Cartesian products of triangle graphs are the Hamminggraphs H ( d , 3 ) {\displaystyle H(d,3)} . Other specific graphs that are...
sum, or bit summation. The Hamming weight is named after Richard Hamming although he did not originate the notion. The Hamming weight of binary numbers...
chemical graph theory. The Fibonacci cube may be defined in terms of Fibonacci codes and Hamming distance, independent sets of vertices in path graphs, or...
instance, the nine-vertex Paley graph (the graph of the 3-3 duoprism) is the Cartesian product of two triangles. The Hamminggraph H ( d , 3 ) {\displaystyle...
graph theory, the Dejter graph is a 6-regular graph with 112 vertices and 336 edges. The Dejter graph is obtained by deleting a copy of the Hamming code...
the vertices of a hypercube, in such a way that distance in the graph equals Hamming distance between the corresponding hypercube vertices. Many important...
correction Gray code Hamming codes Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits Hamming distance: sum number...
total influence can also be defined using the discrete Laplacian of the Hamminggraph, suitably normalized: Inf [ f ] = ⟨ f , L f ⟩ {\displaystyle \operatorname...
connecting pairs of vertices whose Hamming distance is exactly four. The 5-regular Clebsch graph is a strongly regular graph of degree 5 with parameters (...
vertices in the graph is equal to the Hamming distance between their labels. Such a labeling is called a Hamming labeling; it represents an isometric embedding...
The Hamming scheme, named after Richard Hamming, is also known as the hyper-cubic association scheme, and it is the most important example for coding...
the second. Under a reasonable representation of graphs, this is equivalent to the earlier Hamming distance definition (up to possibly a change of constants)...
In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all 2 N {\displaystyle 2^{N}}...
mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. FEC can be...
of a path between two points in a graph The minimum distance of a block code in coding theory, the smallest Hamming distance between any two of its code...
reality. Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Hamming imagines...
that Hamming balls have the smallest vertex boundary among all sets of a given size. Hamming balls are sets that contain all points of Hamming weight...
Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal, and in 1993, the Claude E. Shannon Award. In 1998, he received a...
cutwidth of the graph divided by its maximum degree. The book thickness may be much larger than the queue number: ternary Hamminggraphs have logarithmic...
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