文摘
Let W(G),Sz(G)W(G),Sz(G) and Sz∗(G)Sz∗(G) be the Wiener index, Szeged index and revised Szeged index of a connected graph GG, respectively. Call Ln,rLn,r a lollipop if it is obtained by identifying a vertex of CrCr with an end-vertex of Pn−r+1Pn−r+1. For a connected unicyclic graph GG with n≥4n≥4 vertices, Hansen et al. (2010) conjectured: equation(A)Sz(G)W(G)≤2−8n2+7,if n is odd,2,if n is even,equation(B)Sz∗(G)W(G)≥1+3(n2+4n−6)2(n3−7n+12),if n≤9,1+24(n−2)n3−13n+36,if n≥10,equation(C)Sz∗(G)W(G)≤2+2n2−1,if n is odd,2,if n is even,where the equality in (A) holds if and only if GG is the lollipop Ln,n−1Ln,n−1 if nn is odd, and the cycle CnCn if nn is even; the equality in (B) holds if and only if GG is the lollipop Ln,3Ln,3 if n≤9n≤9, and Ln,4Ln,4 if n≥10n≥10, whereas the equality in (C) holds if and only if GG is the cycle CnCn. In this paper, we not only confirm these conjectures but also determine the lower bound of Sz∗(G)∕W(G)Sz∗(G)∕W(G) (resp. Sz(G)∕W(G)Sz(G)∕W(G)) for cyclic graphs GG. The extremal graphs that achieve these lower bounds are characterized.
