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Algebraic properties of trees by Ladislav Nebeský

By Ladislav Nebeský

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Morris Trees are among the most important graphs. This is mainly due to the fact that many of the applications of graph theory, directly or indirectly, involve trees. 3. We begin this chapter by introducing the concept of a tree and its properties. Many applications of trees will become self-evident once the reader has grasped a few simple notions. Later in the present chapter we go on to introduce various results concerning trees, distances between trees, binary trees, tree enumeration, spanning trees, and fundamental cycles.

1. Our strategy is to show that any tree T has the same centre as a tree T', obtained from it by removing from T all of its pendant vertices. Naturally the eccentricity of any vertex v, in T is equal to the distance from v to some pendant vertex in T. Therefore the eccentricity of each vertex in T' is one less than the eccentricity of that vertex in T. Therefore the vertices of minimum eccentricity in T will have minimum eccentricity in T'. Therefore T and T' have the same centre. Let us successively remove the pendant points all at once from T.

1975). This has been done by Foulds and Robinson (1980, 1981, 1984, 1985, and 1988) . 3. I. The 12 classes of phylogenies. 6 Spanning Trees We shall now consider trees connected graph G = (V, E), T = (V', E'). The edges E', which are not in T are called as subgraphs of larger graphs. Consider a which contains a subgraph which is the tree of T are called branches and the edges of G chords (both relative to T). Definition If V' =V then T is said to be a spanning tree of the graph G. That is, a spanning tree contains all the vertices of the graph of which it is a subgraph.

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