Disk Graph 2 1 15 Multiplication

• 2 Multiplication Chart
• Multiplication 0 1 2 Worksheet
• 3 By 1 Multiplication Worksheet
• Disk Graph 2 1 15 Multiplication Worksheet

Matrix A 3 7 3 6 9 2 0 3 0 2 1 7 2 2 7 9 Matrix B 6 5 5 2 1 7 9 6 6 6 8 9 0 3 5 2 Multiplication of A and B 43 100 132 87 56 68 78 36 8 41 61 35 56 93 129 97 An approach without using global variables: NOTE. It is advised to execute the program in linux based system Compile in linux using following code. Sparse Matrix-Vector Multiplication and Graph Partitioning. Lay out matrix and vectors by rows. y(i) = sum(A(i,j).x(j)). To partition into 2k parts, bisect graph recursively k times. Catalyurek Recursive Bisection.2 1 0 2 -2 -3 1 Nodes tentatively moved (and cut size after each pair).

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In graph theory, the Cartesian productG◻{displaystyle square }H of graphs G and H is a graph such that

• the vertex set of G◻{displaystyle square }H is the Cartesian productV(G) × V(H); and
• two vertices (u,u' ) and (v,v' ) are adjacent in G◻{displaystyle square }H if and only if either
  • u = v and u' is adjacent to v' in H, or
  • u' = v' and u is adjacent to v in G.

The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969].

The operation is associative, as the graphs (F◻{displaystyle square }G) ◻{displaystyle square }H and F◻{displaystyle square } (G◻{displaystyle square }H) are naturally isomorphic.The operation is commutative as an operation on isomorphismclasses of graphs, and more strongly the graphs G◻{displaystyle square }H and H◻{displaystyle square }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.^[1]

Examples[edit]

• The Cartesian product of two edges is a cycle on four vertices: K₂◻{displaystyle square } K₂ = C₄.
• The Cartesian product of K₂ and a path graph is a ladder graph.
• The Cartesian product of two path graphs is a grid graph.
• The Cartesian product of n edges is a hypercube:

(K2)◻n=Qn.{displaystyle (K_{2})^{square n}=Q_{n}.}

Thus, the Cartesian product of two hypercube graphs is another hypercube: Q_i◻{displaystyle square } Q_j = Q_i+j.

• The Cartesian product of two median graphs is another median graph.
• The graph of vertices and edges of an n-prism is the Cartesian product graph K₂◻{displaystyle square } C_n.
• The rook's graph is the Cartesian product of two complete graphs.

Properties[edit]

If 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.^[2] However, 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) ◻{displaystyle square } (K₁ + K₂³) = (K₁ + K₂² + K₂⁴) ◻{displaystyle square } (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.^[3]

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 ◻{displaystyle square } H) = max {χ(G), χ(H)}.^[4]

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 Vizing (1963) showed it satisfies the inequalities

α(G)α(H) + min{|V(G)|-α(G),|V(H)|-α(H)} ≤ α(G◻{displaystyle square }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◻{displaystyle square }H) ≥ γ(G)γ(H).

The Cartesian product of unit distance graphs is another unit distance graph.^[5]

Cartesian product graphs can be recognized efficiently, in linear time.^[6]

Algebraic graph theory[edit]

Algebraic graph theory can be used to analyse the Cartesian graph product.If the graph G1{displaystyle G_{1}} has n1{displaystyle n_{1}} vertices and the n1×n1{displaystyle n_{1}times n_{1}} adjacency matrix A1{displaystyle mathbf {A} _{1}}, and the graph G2{displaystyle G_{2}} has n2{displaystyle n_{2}} vertices and the n2×n2{displaystyle n_{2}times n_{2}} adjacency matrix A2{displaystyle mathbf {A} _{2}}, then the adjacency matrix of the Cartesian product of both graphs is given by

A1◻2=A1⊗In2+In1⊗A2{displaystyle mathbf {A} _{1square 2}=mathbf {A} _{1}otimes mathbf {I} _{n_{2}}+mathbf {I} _{n_{1}}otimes mathbf {A} _{2}},

where ⊗{displaystyle otimes } denotes the Kronecker product of matrices and In{displaystyle mathbf {I} _{n}} denotes the n×n{displaystyle ntimes n}identity matrix.^[7] The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors.

Category theory[edit]

Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetricclosed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs.^[8] The internal hom [G,H]{displaystyle [G,H]} for the cartesian product of graphs has graph homomorphisms from G{displaystyle G} to H{displaystyle H} as vertices and 'unnatural transformations' between them as edges.^[8] Native instruments komplete kontrol 2 3 0.

History[edit]

According to Imrich & Klavžar (2000), Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by Gert Sabidussi (1960).

Notes[edit]

1. ^Hahn & Sabidussi (1997).
2. ^Sabidussi (1960); Vizing (1963).
3. ^Imrich & Klavžar (2000), Theorem 4.19.
4. ^Sabidussi (1957).
5. ^Horvat & Pisanski (2010).
6. ^Imrich & Peterin (2007). For earlier polynomial time algorithms see Feigenbaum, Hershberger & Schäffer (1985) and Aurenhammer, Hagauer & Imrich (1992).
7. ^Kaveh & Rahami (2005).
8. ^ ^abWeber 2013.

References[edit]

• Aurenhammer, F.; Hagauer, J.; Imrich, W. (1992), 'Cartesian graph factorization at logarithmic cost per edge', Computational Complexity, 2 (4): 331–349, doi:10.1007/BF01200428, MR1215316.
• Feigenbaum, Joan; Hershberger, John; Schäffer, Alejandro A. (1985), 'A polynomial time algorithm for finding the prime factors of Cartesian-product graphs', Discrete Applied Mathematics, 12 (2): 123–138, doi:10.1016/0166-218X(85)90066-6, MR0808453.
• Hahn, Geňa; Sabidussi, Gert (1997), Graph symmetry: algebraic methods and applications, NATO Advanced Science Institutes Series, 497, Springer, p. 116, ISBN978-0-7923-4668-5.
• Horvat, Boris; Pisanski, Tomaž (2010), 'Products of unit distance graphs', Discrete Mathematics, 310 (12): 1783–1792, doi:10.1016/j.disc.2009.11.035, MR2610282.
• Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN0-471-37039-8.
• Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Products, A. K. Peters, ISBN1-56881-429-1.
• Imrich, Wilfried; Peterin, Iztok (2007), 'Recognizing Cartesian products in linear time', Discrete Mathematics, 307 (3–5): 472–483, doi:10.1016/j.disc.2005.09.038, MR2287488.
• Kaveh, A.; Rahami, H. (2005), 'A unified method for eigendecomposition of graph products', Communications in Numerical Methods in Engineering with Biomedical Applications, 21 (7): 377–388, doi:10.1002/cnm.753, MR2151527.
• Sabidussi, G. (1957), 'Graphs with given group and given graph-theoretical properties', Canadian Journal of Mathematics, 9: 515–525, doi:10.4153/CJM-1957-060-7, MR0094810.
• Sabidussi, G. (1960), 'Graph multiplication', Mathematische Zeitschrift, 72: 446–457, doi:10.1007/BF01162967, hdl:10338.dmlcz/102459, MR0209177.
• Vizing, V. G. (1963), 'The Cartesian product of graphs', Vycisl. Sistemy, 9: 30–43, MR0209178.
• Weber, Mark (2013), 'Free products of higher operad algebras', TAC, 28 (2): 24–65.

External links[edit]

• Weisstein, Eric W.'Graph Cartesian Product'. MathWorld.

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13 x 5 =

6 x 14 = Onlyoffice 4 0.

15 x 8 =

8 x 20 =

—— End of Page 1 ——

A Cartesian product is vertex transitive if and only if each of its factors is.^[3]

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 ◻{displaystyle square } H) = max {χ(G), χ(H)}.^[4]

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 Vizing (1963) showed it satisfies the inequalities

α(G)α(H) + min{|V(G)|-α(G),|V(H)|-α(H)} ≤ α(G◻{displaystyle square }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◻{displaystyle square }H) ≥ γ(G)γ(H).

The Cartesian product of unit distance graphs is another unit distance graph.^[5]

Cartesian product graphs can be recognized efficiently, in linear time.^[6]

Algebraic graph theory[edit]

Algebraic graph theory can be used to analyse the Cartesian graph product.If the graph G1{displaystyle G_{1}} has n1{displaystyle n_{1}} vertices and the n1×n1{displaystyle n_{1}times n_{1}} adjacency matrix A1{displaystyle mathbf {A} _{1}}, and the graph G2{displaystyle G_{2}} has n2{displaystyle n_{2}} vertices and the n2×n2{displaystyle n_{2}times n_{2}} adjacency matrix A2{displaystyle mathbf {A} _{2}}, then the adjacency matrix of the Cartesian product of both graphs is given by

A1◻2=A1⊗In2+In1⊗A2{displaystyle mathbf {A} _{1square 2}=mathbf {A} _{1}otimes mathbf {I} _{n_{2}}+mathbf {I} _{n_{1}}otimes mathbf {A} _{2}},

where ⊗{displaystyle otimes } denotes the Kronecker product of matrices and In{displaystyle mathbf {I} _{n}} denotes the n×n{displaystyle ntimes n}identity matrix.^[7] The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors.

Category theory[edit]

Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetricclosed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs.^[8] The internal hom [G,H]{displaystyle [G,H]} for the cartesian product of graphs has graph homomorphisms from G{displaystyle G} to H{displaystyle H} as vertices and 'unnatural transformations' between them as edges.^[8] Native instruments komplete kontrol 2 3 0.

History[edit]

According to Imrich & Klavžar (2000), Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by Gert Sabidussi (1960).

Notes[edit]

1. ^Hahn & Sabidussi (1997).
2. ^Sabidussi (1960); Vizing (1963).
3. ^Imrich & Klavžar (2000), Theorem 4.19.
4. ^Sabidussi (1957).
5. ^Horvat & Pisanski (2010).
6. ^Imrich & Peterin (2007). For earlier polynomial time algorithms see Feigenbaum, Hershberger & Schäffer (1985) and Aurenhammer, Hagauer & Imrich (1992).
7. ^Kaveh & Rahami (2005).
8. ^ ^abWeber 2013.

References[edit]

• Aurenhammer, F.; Hagauer, J.; Imrich, W. (1992), 'Cartesian graph factorization at logarithmic cost per edge', Computational Complexity, 2 (4): 331–349, doi:10.1007/BF01200428, MR1215316.
• Feigenbaum, Joan; Hershberger, John; Schäffer, Alejandro A. (1985), 'A polynomial time algorithm for finding the prime factors of Cartesian-product graphs', Discrete Applied Mathematics, 12 (2): 123–138, doi:10.1016/0166-218X(85)90066-6, MR0808453.
• Hahn, Geňa; Sabidussi, Gert (1997), Graph symmetry: algebraic methods and applications, NATO Advanced Science Institutes Series, 497, Springer, p. 116, ISBN978-0-7923-4668-5.
• Horvat, Boris; Pisanski, Tomaž (2010), 'Products of unit distance graphs', Discrete Mathematics, 310 (12): 1783–1792, doi:10.1016/j.disc.2009.11.035, MR2610282.
• Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN0-471-37039-8.
• Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Products, A. K. Peters, ISBN1-56881-429-1.
• Imrich, Wilfried; Peterin, Iztok (2007), 'Recognizing Cartesian products in linear time', Discrete Mathematics, 307 (3–5): 472–483, doi:10.1016/j.disc.2005.09.038, MR2287488.
• Kaveh, A.; Rahami, H. (2005), 'A unified method for eigendecomposition of graph products', Communications in Numerical Methods in Engineering with Biomedical Applications, 21 (7): 377–388, doi:10.1002/cnm.753, MR2151527.
• Sabidussi, G. (1957), 'Graphs with given group and given graph-theoretical properties', Canadian Journal of Mathematics, 9: 515–525, doi:10.4153/CJM-1957-060-7, MR0094810.
• Sabidussi, G. (1960), 'Graph multiplication', Mathematische Zeitschrift, 72: 446–457, doi:10.1007/BF01162967, hdl:10338.dmlcz/102459, MR0209177.
• Vizing, V. G. (1963), 'The Cartesian product of graphs', Vycisl. Sistemy, 9: 30–43, MR0209178.
• Weber, Mark (2013), 'Free products of higher operad algebras', TAC, 28 (2): 24–65.

External links[edit]

• Weisstein, Eric W.'Graph Cartesian Product'. MathWorld.

—— Note: The Information above this point will not be sent to your printer ——–

13 x 5 =

6 x 14 = Onlyoffice 4 0.

15 x 8 =

8 x 20 =

—— End of Page 1 ——

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2 Multiplication Chart

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Multiplication 0 1 2 Worksheet

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Related Resources

The Multiplication Flashcard shown above is aligned with standard 4NBT05 taken from the Common Core Standards For Mathematics (see the extract below) as are the resources listed below.

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Example/Guidance

Flashcard

• Products to 144 e.g. 12 x 11

Worksheet

• 2 x 1 digit e.g. 54 x 5 requires only 2x, 5x, or 10x multiplication facts.
• 3 x 1 digit e.g. 512 x 7 requires only 2x, 5x, or 10x multiplication facts.
• Creative Multiplication (6-page with guidance notes and examples)
• 2 x 1-digit – Products to 99 e.g. 23 x 2
• 2 x 1-digit – Products to 999 e.g. 93 x 4
• 2 x 1-digit – regrouping e.g. 73 x 7
• 3 x 1-digit e.g. 436 x 7
• 2 x 2-digit no regrouping e.g. 42 x 23
• 2 x 2-digit regrouping e.g. 57 x 34
• 4 x 1-digit e.g. 5316 x 8

Third/ Fourth Grade

• Addition & Subtraction Worksheet (Challenging) (4 pages)

Worksheet Generator

• Box Method Multiplication : 1-digit x 2-digit

Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:

3 By 1 Multiplication Worksheet

Use place value understanding and properties of operations to perform multi-digit arithmetic

Disk Graph 2 1 15 Multiplication Worksheet

• Divisibility Rules For 2, 3 ,4, 5 ,6 & 9 (From Example/Guidance)
• Addition & Subtraction (larger numbers) e.g. up to 999,999 (From Worksheet)
• Addition with up to 8 Numbers: Worksheet Generator (From Worksheet)
• Three 3-digit numbers e.g. 247 + 114 + 598 (From Worksheet)
• 4-digit numbers e.g. 3286 – 1468 (From Worksheet)
• Four 4-digit numbers e.g. 2347 + 1214 + 5198 + 6548 (From Worksheet)
• Dividends to 99 with remainders e.g. 97 ÷ 5 (From Worksheet)
• 3-digit by 1-digit e.g. 507 ÷ 9 (From Worksheet)
• 4-digit by 1-digit e.g. 9337 ÷ 6 (From Worksheet)
• 5-digit numbers e.g. 47324 + 24743 + 3255 (From Worksheet)