Discussiones Mathematicae Graph Theory 32(1) (2012) 19-29
doi: 10.7151/dmgt.1582

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Median of a Graph with Respect to Edges

A.P. Santhakumaran

Department of Mathematics
St.Xavier's College (Autonomous)
Palayamkottai - 627 002, India.

Abstract

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 ∈ Vd(v,u), the vertex-to-edge distance sum d1(v) of v is d1(v) = ∑e ∈ Ed(v,e), the edge-to-vertex distance sum d2(e) of e is d2(e) = ∑v ∈ Vd(e,v) and the edge-to-edge distance sum d3(e) of e is d3(e) = ∑f ∈ Ed(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.

Keywords: median, vertex-to-edge median, edge-to-vertex median, edge-to-edge median

2010 Mathematics Subject Classification: 05C12.

References

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Received 4 June 2010
Revised 25 December 2010
Accepted 27 December 2010