Median of a graph with respect to edges
A.P. Santhakumaran
Discussiones Mathematicae Graph Theory, Tome 32 (2012), p. 19-29 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:270947 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1582,
     author = {A.P. Santhakumaran},
     title = {Median of a graph with respect to edges},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {32},
     year = {2012},
     pages = {19-29},
     zbl = {1255.05067},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1582}
}

A.P. Santhakumaran. Median of a graph with respect to edges. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 19-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1582/

[000] [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading MA, 1990). | Zbl 0688.05017

[001] [2] F. Buckley, Z. Miller and P.J. Slater, On graphs containing a given graph as center, J. Graph Theory 5 (1981) 427-434, doi: 10.1002/jgt.3190050413. | Zbl 0449.05056

[002] [3] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill, New Delhi, 2006). | Zbl 1096.05001

[003] [4] L.C. Freeman, Centrality in Social networks; 1. Conceptual clarification, Social Networks 1 (1978/79) 215-239, doi: 10.1016/0378-8733(78)90021-7.

[004] [5] C. Jordan, Sur les assemblages des lignas, J. Reine Angew. Math. 70 (1869) 185-190, doi: 10.1515/crll.1869.70.185.

[005] [6] A.P. Santhakumaran, Center of a graph with respect to edges, SCIENTIA, Series A: Mathematical Sciences 19 (2010) 13-23. | Zbl 1244.05081

[006] [7] P.J. Slater, Some definitions of central structures, preprint. | Zbl 0545.05039

[007] [8] P.J. Slater, Centrality of paths and vertices in a graph : Cores and Pits, Theory and Applications of Graphs, ed, Gary Chartrand, (John Wiley, 1981) 529-542.

[008] [9] B. Zelinka, Medians and Peripherians of trees, Arch. Math., Brno (1968) 87-95. | Zbl 0206.26105