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· 136-140 · 141-145 · 146-150 · 151-155 · 156-160 · 161-165 · 166-170 · 171-175 · 176-180 · 181-185 · 186-190 ·Count the graphs that have vertex set V = {1, 2, 3, ..., n}.
The problem is to let V = {1, 2, 3, ..., n}, and to determine the number of different graphs that can be formed with V as vertex set. See attached file for full problem description.
Subject:
Math
Topic:
Discrete Structures
Posting ID:
28047
OTA ID:
104146
What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphism of G to H. We can think of f as renaming the vertices of G with the names of the vertices of H in a way that preserves adjacency. Less formally, isomorphic graphs have the same drawing (except for the names of the vertices). Do the following: (a) Prove that isomorphic graphs have the same number of vertices. (b) Prove that if f:V(G) -> V(H) is an isomorph... click for more
Subject:
Math
Topic:
Discrete Structures
Posting ID:
28048
OTA ID:
104146
Find the values of alpha and omega for the two graphs given in the attached file (45.4.doc).
The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G. The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of vertices in S are connected by an edge of G. Two graphs, G and H, are given as figures in an attached .doc file (45.4.doc). Find the values of alpha(G), omega(G), alpha(H), and omega(H).
Subject:
Math
Topic:
Discrete Structures
Posting ID:
28050
OTA ID:
104146
The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G. The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of vertices in S are connected by an edge of G. Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following: (a) alpha(G) <= alpha(H) (b) alpha(G) >= alpha(H) (c) omega(G) <= omega(H) (d) omega(G) >= omega(H)
Subject:
Math
Topic:
Discrete Structures
Posting ID:
28051
OTA ID:
104146
Let G be a graph. Then G = (V, E), where V and E are the vertex set and edge set, respectively, of G. The complement of G, which we will refer to as “G bar,” is the graph (V, E bar), where V is the vertex set of G bar (i.e., the vertex set of G bar is identical to the vertex set of G) and E bar is the edge set of G bar. The edge set E bar is defined as follows: For distinct vertices v1, v2, there is an edge that connects v1 and v2 in G bar if and only if there is no edge that connects v1 and v2 in G. Definition: A graph G is self-complementary if G is isomorphic to G bar. Do the following: (a) Show that the graph G = ({a, b, c, d}, {ab, bc, cd}) is self-complementary. (b) F... click for more
Subject:
Math
Topic:
Discrete Structures
Posting ID:
28052
OTA ID:
104146
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