Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond

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