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- W2090029486 abstract "Abstract The Wiener indexW(G) of a graphGis the sumof the distances betweenall pairs of vertices in the graph. The Szeged index Sz(G) of a graph Gis defined as Sz(G) =P e=uv∈E n u (e)n v (e) where n u (e) and n v (e) are,respectively, the number of vertices of Glying closer to vertex uthan tovertex v and the number of vertices of Glying closer to vertex v thanto vertex u. Hansen used the computer programm AutoGraphiX andmade the following conjecture about the Szeged index and the Wienerindex for a bipartite connected graph Gwith n≥ 4 vertices and m≥ nedges:Sz(G) −W(G) ≥ 4n−8.Moreover the bound is best possible as shown by the graph composed ofa cycle on 4 vertices C 4 and a tree T on n− 3 vertices sharing a singlevertex. This paper is to give a confirmative proof to this conjecture. Keywords: Wiener index, Szeged index, bipartite graph.AMS subject classification 2010: 05C12, 05C35, 05C90, 92E10. 1 Introduction All graphs considered in this paper are finite, undirected and simple. We refer thereaders to [2] for terminology and notation. Let G be a connected graph with vertexset V and edge set E. For u,v ∈ V, d(u,v) denotes the distancebetween u and v. TheWienerindexof G is defined asW(G) =X" @default.
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- W2090029486 date "2012-10-24" @default.
- W2090029486 modified "2023-09-27" @default.
- W2090029486 title "On a relation between the Szeged index and the Wiener index for bipartite graphs" @default.
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