词条 | Cartesian product of graphs |
释义 |
In graph theory, the Cartesian product G H of graphs G and H is a graph such that
The operation is associative, as the graphs (F G) H and F (G H) are naturally isomorphic. The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs G H and H G are naturally isomorphic, but it is not commutative as an operation on labeled graphs. The notation G × H has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges.{{sfnp|Hahn|Sabidussi|1997}} Examples
Thus, the Cartesian product of two hypercube graphs is another hypercube: Qi Qj = Qi+j.
PropertiesIf a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs.[1] However, {{harvtxt|Imrich|Klavžar|2000}} describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs: (K1 + K2 + K22) (K1 + K23) = (K1 + K22 + K24) (K1 + K2), where the plus sign denotes disjoint union and the superscripts denote exponentiation over Cartesian products. A Cartesian product is vertex transitive if and only if each of its factors is.[2] A Cartesian product is bipartite if and only if each of its factors is. More generally, the chromatic number of the Cartesian product satisfies the equation χ(G H) = max {χ(G), χ(H)}.{{sfnp|Sabidussi|1957}} The Hedetniemi conjecture states a related equality for the tensor product of graphs. The independence number of a Cartesian product is not so easily calculated, but as {{harvtxt|Vizing|1963}} showed it satisfies the inequalities α(G)α(H) + min{|V(G)|-α(G),|V(H)|-α(H)} ≤ α(G H) ≤ min{α(G) |V(H)|, α(H) |V(G)|}. The Vizing conjecture states that the domination number of a Cartesian product satisfies the inequality γ(G H) ≥ γ(G)γ(H). The Cartesian product of unit distance graphs is another unit distance graph.{{sfnp|Horvat|Pisanski|2010}} Cartesian product graphs can be recognized efficiently, in linear time.[3] Algebraic graph theoryAlgebraic graph theory can be used to analyse the Cartesian graph product. If the graph has vertices and the adjacency matrix , and the graph has vertices and the adjacency matrix , then the adjacency matrix of the Cartesian product of both graphs is given by , where denotes the Kronecker product of matrices and denotes the identity matrix.{{sfnp|Kaveh|Rahami|2005}} The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors. HistoryAccording to {{harvtxt|Imrich|Klavžar|2000}}, Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by {{harvs|first=Gert|last=Sabidussi|authorlink=Gert Sabidussi|year=1960|txt}}. Notes1. ^{{harvtxt|Sabidussi|1960}}; {{harvtxt|Vizing|1963}}. 2. ^{{harvtxt|Imrich|Klavžar|2000}}, Theorem 4.19. 3. ^{{harvtxt|Imrich|Peterin|2007}}. For earlier polynomial time algorithms see {{harvtxt|Feigenbaum|Hershberger|Schäffer|1985}} and {{harvtxt|Aurenhammer|Hagauer|Imrich|1992}}. References
| last1 = Aurenhammer | first1 = F. | author1-link = Franz Aurenhammer | last2 = Hagauer | first2 = J. | last3 = Imrich | first3 = W. | doi = 10.1007/BF01200428 | issue = 4 | journal = Computational Complexity | mr = 1215316 | pages = 331–349 | title = Cartesian graph factorization at logarithmic cost per edge | volume = 2 | year = 1992}}.
| last1 = Feigenbaum | first1 = Joan | author1-link = Joan Feigenbaum | last2 = Hershberger | first2 = John | author2-link = John Hershberger | last3 = Schäffer | first3 = Alejandro A. | doi = 10.1016/0166-218X(85)90066-6 | issue = 2 | journal = Discrete Applied Mathematics | mr = 808453 | pages = 123–138 | title = A polynomial time algorithm for finding the prime factors of Cartesian-product graphs | volume = 12 | year = 1985}}.
| last1 = Hahn | first1 = Geňa | last2 = Sabidussi | first2 = Gert | author2-link = Gert Sabidussi | isbn = 978-0-7923-4668-5 | page = 116 | publisher = Springer | series = NATO Advanced Science Institutes Series | title = Graph symmetry: algebraic methods and applications | url = https://books.google.com/books?id=-tIaXdII8egC&pg=PA116 | volume = 497 | year = 1997}}.
| last1 = Horvat | first1 = Boris | last2 = Pisanski | first2 = Tomaž | author2-link = Tomaž Pisanski | doi = 10.1016/j.disc.2009.11.035 | issue = 12 | journal = Discrete Mathematics | mr = 2610282 | pages = 1783–1792 | title = Products of unit distance graphs | volume = 310 | year = 2010}}.
| last1 = Imrich | first1 = Wilfried |authorlink = Wilfried Imrich | last2 = Klavžar | first2 = Sandi | isbn = 0-471-37039-8 | publisher = Wiley | title = Product Graphs: Structure and Recognition | year = 2000}}.
| last1 = Imrich | first1 = Wilfried |authorlink = Wilfried Imrich | last2 = Klavžar | first2 = Sandi | last3 = Rall | first3 = Douglas F. | isbn = 1-56881-429-1 | publisher = A. K. Peters | title = Graphs and their Cartesian Products | year = 2008}}.
| last1 = Imrich | first1 = Wilfried | authorlink = Wilfried Imrich | last2 = Peterin | first2 = Iztok | doi = 10.1016/j.disc.2005.09.038 | issue = 3-5 | journal = Discrete Mathematics | mr = 2287488 | pages = 472–483 | title = Recognizing Cartesian products in linear time | volume = 307 | year = 2007}}.
| last1 = Kaveh | first1 = A. | last2 = Rahami | first2 = H. | doi = 10.1002/cnm.753 | issue = 7 | journal = Communications in Numerical Methods in Engineering with Biomedical Applications | mr = 2151527 | pages = 377–388 | title = A unified method for eigendecomposition of graph products | volume = 21 | year = 2005}}.
| last = Sabidussi | first = G. | authorlink = Gert Sabidussi | doi = 10.4153/CJM-1957-060-7 | journal = Canadian Journal of Mathematics | mr = 0094810 | pages = 515–525 | title = Graphs with given group and given graph-theoretical properties | volume = 9 | year = 1957}}.
| last = Sabidussi | first = G. | authorlink = Gert Sabidussi | doi = 10.1007/BF01162967 | journal = Mathematische Zeitschrift | mr = 0209177 | pages = 446–457 | title = Graph multiplication | volume = 72 | year = 1960}}.
| last = Vizing | first = V. G. | author-link = Vadim G. Vizing | journal = Vycisl. Sistemy | mr = 0209178 | pages = 30–43 | title = The Cartesian product of graphs | volume = 9 | year = 1963}}. External links
1 : Graph products |
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