منابع مشابه
Realizing the distance matrix of a graph
An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntain...
متن کاملA constructive algorithm for realizing a distance matrix
The natural metric of a weighted graph is the length of the shortest paths between all pairs of vertices. The investigated problem consists in a representation of a given metric by a graph, such that the total length of the graph is minimized. For that purpose, we give a constructive algorithm based on a technique of reduction, fusion and deletion. We then show some results on a set of various ...
متن کاملProduct Distance Matrix of a Graph and Squared Distance Matrix of a Tree
Let G be a strongly connected, weighted directed graph. We define a product distance η(i, j) for pairs i, j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained....
متن کاملOn the determinant of q-distance matrix of a graph
In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...
متن کاملInverse of the distance matrix of a block graph
Graham et al. [3] proved a very attractive theorem about the determinant of the distance matrix DG of a connected graph G as a function of the distance matrix of its 2-connected blocks. In a connected graph, the distance between two vertices d(u, v) is the length of the shortest path between them. Let A be an n n matrix. Recall that for 1 i, j n, the cofactor ci,j is defined as ( 1) iþj times t...
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ژورنال
عنوان ژورنال: Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics
سال: 1966
ISSN: 0022-4340
DOI: 10.6028/jres.070b.013