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Adjacency Matrices Two type of matrices: - Based on the adjacency of vertices Graph Theory - Based on incidence of vertices and edges 8.5. Representation of Graphs Definition - Suppose that G = (V, E) is a simple graph where |V| = n. Suppose that the vertices of G are v1, v2, Muhammad Arief …, vn The adjacency matrix A of G (or AG) is the download dari http://arief.ismy.web.id n x n zero-one matrix with 1 as its (i, j)th entry when vi and vj, are adjacent and 0 as its (i, j)th entry when they are not adjacent . http://arief.ismy.web.id http://arief.ismy.web.id Adjacency Matrices Example Adjacency Matrix? A = [aij] aij = 1 if {vi, vj} is an edge of G =0 otherwise http://arief.ismy.web.id http://arief.ismy.web.id Example Example The Graph? Adjacency Matrix? http://arief.ismy.web.id http://arief.ismy.web.id Incidence Matrices Example Definition - Suppose that G = (V, E) is a simple graph where Incidence Matrix? |V| = n. Suppose that the vertices of G are v1, v2, …, vn and the edges of G are e1, e2, …, em. The incidence matrix is the n x m matrix M = [mij], where: mij =1 when edge ej is incidence with vi =0 otherwise http://arief.ismy.web.id http://arief.ismy.web.id Example Incidence Matrix? Graph Theory 8.6. Isomorphisms of Graphs Muhammad Arief download dari http://arief.ismy.web.id http://arief.ismy.web.id http://arief.ismy.web.id One-to-One Function Onto Function Definition: Definition: A function f is said to be one-to-one, or A function f from A to B is called onto, or injective, if and only if f(a) = f(b) implies that surjective, if and only if for every elements a = b for all a and b in the domain of f. A b ∈ B there is an element a ∈ A with f(a) = function is said to be an injection if it is one- b. A function f is called a surjection if it is to-one. onto. Notation: ∀a∀b (f(a) = f(b) → a = b) Notation: ∀a∀b (a ≠ b → f(a) ≠ f(b)) ∀y∃x ( f(x) = y) http://arief.ismy.web.id http://arief.ismy.web.id One-to-One Correspondence One-to-One Correspondence Definition: The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. A = {a, b, c, d} B = {1, 2, 3, 4} f(a) = 4, f(b) = 2, f(c) = 1, f(d) = 3. One-to-one, onto Is this a bijection? Yes, because it is both one-to-one and onto. http://arief.ismy.web.id http://arief.ismy.web.id One-to-One Correspondence Isomorphisme of Graphs Definition - The simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2, for all a and b in V1. Such a function f is called an isomorphism. One-to-one, not onto Onto, not one-to-one One-to-one, onto Neither one-to-one, nor onto Not a function http://arief.ismy.web.id http://arief.ismy.web.id Example Example Show that G = (V, E) and The function f with f(u1) = v1, f(u2) = v4, f(u3) = v3 H = (W, F) are and f(u4) = v2 is a one-to-one correspondence isomorphic between G and H. For every adjacent vertices in G, there is also corresponding adjacent vertices in H. http://arief.ismy.web.id http://arief.ismy.web.id What to check? Example Have the same number of vertices. Determine whether the graphs shown below are isomorphic. Have the same number of edges OR have the same number of the degree of the vertices. Have the same number of the degree of the adjacent vertices. Have the same adjacency matrices. http://arief.ismy.web.id http://arief.ismy.web.id Example Example Determine whether the graphs shown below are Determine whether the graphs shown below are isomorphic. isomorphic. http://arief.ismy.web.id http://arief.ismy.web.id Example http://arief.ismy.web.id

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posted: | 5/1/2012 |

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