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Trees and Distance

2.1 Basic properties. Acyclic : a graph with no cycleForest : acyclic graphTree : connected acyclic graph Leaf : a vertex of degree 1Spanning subgraph of G : a subgraph with vertex set V(G)Spanning tree : a spanning subgraph that is a treeStar : a tree consisting of one vertex adjacent to all

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Trees and Distance

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    1. Trees and Distance

    2. 2.1 Basic properties Acyclic : a graph with no cycle Forest : acyclic graph Tree : connected acyclic graph Leaf : a vertex of degree 1 Spanning subgraph of G : a subgraph with vertex set V(G) Spanning tree : a spanning subgraph that is a tree Star : a tree consisting of one vertex adjacent to all the others.

    3. Properties of trees Lemma : every tree with at least two vertices has at least two leaves. Deleting a leaf from an n-vertex tree produces a tree with n-1 vertices. Theorem : (A) G is connected and has no cycles (B) G is connected and has n-1 edges (C) G has n-1 edges and no cycles (D) For u,v ?V(G), G has exactly one u,v-path

    4. Proof theorem A -> B,C : by induction on n. B-> A,C : delete edges from cycles of G one by one. C -> A,B : let G has k components. e(Gi) = n(Gi)-1 . e(G)=?i[n(Gi)-1]=n-k. A->D : if some pair of vertices is connected by more than one path, it will form a cycle. D->A : if G has a cycle, then G has more than one u,v path .

    5. Properties of trees Corollary : (a) every edge of a tree is a cut edge. (b) adding one edge to a tree forms exactly one cycle. (c) every connected graph contains a spanning tree. Proposition : if T,T are spanning trees of G and e? E(T)-E(T), then there is an edge e? E(T)-E(T) such that T-e+e is a spanning tree of G.

    6. Properties of trees Proposition : if T is a tree with k edges and G is a simple graph with ?(G)?k, then T is a subgraph of G.

    7. Distance in trees and graphs d(u,v) : is the least length of a u,v-path Diameter : max u,v?V(G) d(u,v) Eccentricity of a vertex u, ?(u) : max u,v?V(G) d(u,v) Radius of G : is min u?V(G) ?(u)

    8. Distance in trees and graphs Theorem : if G is a simple graph, then diamG?3 -> diamG?3 Proof : ?u,v have no common neighbor. x?V(G)-{u,v} has at least one of {u,v} as a nonneighbor. This makes x adjacent in G to at least one of {u,v}. uv?E(G).

    9. Distance in trees and graphs Center of G : the subgraph induced by the vertices of minimum eccentricity. Theorem : the center of a tree is a vertex or an edge.

    10. Wiener index Wiener index of G : D(G) = ?u,v?V(G)dG(u,v) Theorem : among trees with n vertices, the Wiener index D(T) is minimized by stars and maximized by paths ,both uniquely.

    11. 2.2 spanning trees and enumeration Prfer code :

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