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

Let W ( G ) , S z ( G ) and S z ź ( G ) be the Wiener index, Szeged index and revised Szeged index of a connected graph G , respectively. Call L n , r a lollipop if it is obtained by identifying a vertex of C r with an end-vertex of P n - r + 1 . For a connected unicyclic graph G with n ź 4 vertices, Hansen et al.ź(2010) conjectured: (A) S z ( G ) W ( G ) ź 2 - 8 n 2 + 7 , ifź n źisźodd, 2 , ifź n źisźeven, (B) S z ź ( G ) W ( G ) ź 1 + 3 ( n 2 + 4 n - 6 ) 2 ( n 3 - 7 n + 12 ) , ifź n ź 9 , 1 + 24 ( n - 2 ) n 3 - 13 n + 36 , ifź n ź 10 , (C) S z ź ( G ) W ( G ) ź 2 + 2 n 2 - 1 , ifź n źisźodd, 2 , ifź n źisźeven, where the equality in (A) holds if and only if G is the lollipop L n , n - 1 if n is odd, and the cycle C n if n is even; the equality in (B) holds if and only if G is the lollipop L n , 3 if n ź 9 , and L n , 4 if n ź 10 , whereas the equality in (C) holds if and only if G is the cycle C n . In this paper, we not only confirm these conjectures but also determine the lower bound of S z ź ( G ) ź W ( G ) (resp. S z ( G ) ź W ( G ) ) for cyclic graphs G . The extremal graphs that achieve these lower bounds are characterized.