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MC 302 Graph Theory Tuesday, 9

Regular Graphs. A regular graph is one in which all vertices have the same degree.Examples:Is Kn regular? Is Km, n regular?If a graph with n vertices is k-regular, how many edges does it have?If two graphs on n vertices are both k-regular for some k, must they be isomorphic?For each k = 0, ...

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MC 302 Graph Theory Tuesday, 9

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    1. MC 302 Graph Theory Tuesday, 9/14/04 Textbook: A Friendly Introduction to Graph Theory, by Fred Buckley and Marty Lewinter, Prentice Hall Revised reading & exercises for 9/9/04 class: Read 2.1-2.2. Review Chap. 1 as needed. Exercises: p. 55, 2.1: #1-3; p. 62, 2.2: #5 Todays reading & exercises: Read 2.2-2.3 (Well finish 2.3 material next time) Exercises: p. 62, 2.2: #1,6,7,8; p. 72, 2.3:#4,5,16 Today: (We always start with questions!) Cool graph website: http://www.touchgraph.com Worksheet #1 Identical and Isomorphic Graphs Regular graphs, subgraphs, unions and intersections HAND-IN HOMEWORK #1: Due Monday at 5 PM.

    2. Regular Graphs A regular graph is one in which all vertices have the same degree. Examples: Is Kn regular? Is Km, n regular? If a graph with n vertices is k-regular, how many edges does it have? If two graphs on n vertices are both k-regular for some k, must they be isomorphic? For each k = 0, ..., 4, draw a k-regular graph on 5 vertices, or explain why one cannot exist.

    3. New Graphs from Old: Subgraphs H is a subgraph of G if V(H) is a subset of V(G) and E(H) is a subset of E(G). We usually also call H a subgraph of G if H is isomorphic to a subgraph of G. A spanning subgraph of G is a subgraph H containing all the vertices of G.

    4. Induced Subgraphs Given a subset V' of V(G), the induced subgraph on V' is the subgraph H with V(H) = V', and with E(H) containing all possible edges of G between vertices of V' Given a subset E' of E(G), the edge-induced subgraph on E' is the subgraph having E(H) = E' and V(H) containing all the vertices incident to edges in E'

    5. Subgraph examples For the graph G to the right, find a graph on 4 vertices that is a subgraph of G a graph on 4 vertices that is not a subgraph of G a spanning subgraph of G a minimal spanning subgraph of G the induced subgraph on V' = {1, 2, 4, 5} a subgraph of G that is not induced The edge-induced subgraph on E' = {a, c, e, g}

    6. Subgraphs Formed by Deletions For v ? V, G - v (or G \ v)is the induced graph on V - {v} For e ? E, G - e is the edge-induced graph on E - {e} For V' ? V, G - V' is the induced graph on V - V' For E' ? V, G - E' is the edge-induced graph on E - E'

    7. Unions and Intersections If G1 and G2 are two subgraphs of G, then G1 ? G2 has vertex set V(G1) ? V(G2) and edge set E(G1) ? E(G2) If G1 and G2 are two subgraphs of G with at least one vertex in common, then G1 ? G2 has vertex set V(G1) ? V(G2) and edge set E(G1) ? E(G2) Remember that a graph is not permitted to have zero vertices -- hence the extra requirement

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