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2012•Transactions on combinatoricsOpen access

On a relation between Szeged and Wiener indices of bipartite graphs

Lung‐Chi Chen, Xueliang Li, Mengmeng Liu, İvan Gutman

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Abstract

Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index $Sz(G)$ and the‎ ‎Wiener index $W(G)$ of a connected bipartite graph $G$ with $n geq‎ ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$Sz(G)-W(G) geq 4n-8$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎.

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What this paper is about

Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index $Sz(G)$ and the‎ ‎Wiener index $W(G)$ of a connected bipartite graph $G$ with $n geq‎ ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$Sz(G)-W(G) geq 4n-8$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎.

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Available abstract

Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index $Sz(G)$ and the‎ ‎Wiener index $W(G)$ of a connected bipartite graph $G$ with $n geq‎ ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$Sz(G)-W(G) geq 4n-8$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎.

Key concepts: Wiener index, Mathematics, Bipartite graph, Combinatorics, Conjecture, Index (typography), Graph, Relation (database)

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