2011/04/12 by Xueliang Li, Mengmeng Liu, Li, Xueliang +1 · 1 citation
Mathematics · #05C12 #05C35 #05C90 #92E10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C12 #msc:05C35 #msc:05C90 #msc:92E10
paper · pdf · doi:10.48550/arxiv.1104.2122
7 pages
arxiv created 2011/04/12 · arxiv updated 2011/04/13
The revised Szeged index Sz^*(G) is defined as Sz^*(G)=∑e=uv ∈ E(nu(e)+ n0(e)/2)(nv(e)+ n0(e)/2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. Hansen used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph G of order n ≥ 6: Sz^*(G)≤ \arrayll (n3+n2-n-1)/4, if n is odd, (n3+n2-n)/4, if n is even. array. with equality if and only if G is the graph obtained from the cycle Cn-1 by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.