Median of a graph with respect to edges

• Autorzy: A.P. Santhakumaran
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

median; vertex-to-edge median; edge-to-vertex median; edge-to-edge median

Kategorie tematyczne

• 05C12: Distance in graphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2012
• Tom: 32
• Numer: 1
• Strony: 19-29

Daty

wydano

2012

otrzymano

2010-06-04

poprawiono

2010-12-25

zaakceptowano

2010-12-27

Twórcy

autor

A.P. Santhakumaran

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

Bibliografia

• [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading MA, 1990).
• [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.
• [3] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill, New Delhi, 2006).
• [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.
• [5] C. Jordan, Sur les assemblages des lignas, J. Reine Angew. Math. 70 (1869) 185-190, doi: 10.1515/crll.1869.70.185.
• [6] A.P. Santhakumaran, Center of a graph with respect to edges, SCIENTIA, Series A: Mathematical Sciences 19 (2010) 13-23.
• [7] P.J. Slater, Some definitions of central structures, preprint.
• [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.
• [9] B. Zelinka, Medians and Peripherians of trees, Arch. Math., Brno (1968) 87-95.

Identyfikatory

DOI

10.7151/dmgt.1582