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Median of a graph with respect to edges

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Santhakumaran A.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 1

19-29

For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is $d(v) = ∑_{u ∈ V}d(v,u)$, the vertex-to-edge distance sum d₁(v) of v is $d₁(v) = ∑_{e ∈ E}d(v,e)$, the edge-to-vertex distance sum d₂(e) of e is $d₂(e) = ∑_{v ∈ V}d(e,v)$ and the edge-to-edge distance sum d₃(e) of e is $d₃(e) = ∑_{f ∈ E}d(e,f)$. The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set M₁(G) of all vertices v for which d₁(v) is minimum is the vertex-to-edge median of G; the set M₂(G) of all edges e for which d₂(e) is minimum is the edge-to-vertex median of G; and the set M₃(G) of all edges e for which d₃(e) is minimum is the edge-to-edge median of G. We determine these medians for some classes of graphs. We prove that the edge-to-edge median of a graph is the same as the median of its line graph. It is shown that the center and the median; the vertex-to-edge center and the vertex-to-edge median; the edge-to-vertex center and the edge-to-vertex median; and the edge-to-edge center and the edge-to-edge median of a graph are not only different but can be arbitrarily far apart.

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1 Discussiones Mathematicae Graph Theory

1 Santhakumaran A.

1 2012

Wyniki wyszukiwania

Median of a graph with respect to edges