# Distance Matrix of a Tree

Chapter
Part of the Universitext book series (UTX)

Let G be a connected graph with V(G) = {1,⋯n}. Recall that the distance d(i, j) between the vertices i and j of G is the length of a shortest path from i to j. The distance matrix D(G) of G is an n×n matrix with its rows and columns indexed by V(G). For ij, the (i, j)-entry di j of G is set equal to d(i, j): Also, d ii = 0, i = 1,⋯n. We will often denote D(G) simply by D. Clearly, D is a symmetric matrix with zeros on the diagonal. The distance, as a function on V(GV(G), satisfies the triangle inequality. Thus, for any vertices i, j and k.

## Keywords

Distance Matrix Connected Graph Algebraic Multiplicity Unicyclic Graph Pendant Vertex
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