Continuous Mean Distance of a Weighted Graph

On weighted mean distance

In this paper we study the average distance in weighted graphs. More precisely, we consider assignments of families of non-negative weights to the edges. The aim is to maximise (minimise, respectively) the average distance in the resulting weighted graph. Two variants of the problem are considered depending on whether the collection of weights is fixed or not. The main results of this paper are...

A Method for Defuzzication by Weighted Distance

Different-Distance Sets in a Graph

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

On the Mean Distance between Points O. a Graph

Given any graph G with vertex set V(G), let r(G) _~'-1-, where deg(x) is the degree of x in G. x V(G) de g (x) Further, let a(G) denote the mean distance between vertices of G, i .e ., u(G) =. F .-d(x,y) I IV(T ,) where d(x,y) is the length of the shortest path connecting x and y in G, and the sum is taken over all pairs {x,y} of distinct vertices. The following attractive conjecture was made b...

Stipple Placement using Distance in a Weighted Graph

We present a stipple placement method which provides extra emphasis to image features, especially edges. Our algorithm transforms an image into a regular graph, with edge weights given by the local gradient magnitude of the input image. Then, a variant of Disjkstra’s algorithm is used to place stipples: new stipples are placed along the frontier, which tends to be aligned with image edges. The ...