Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching

AbstractLet T be an acyclic graph without perfect matching and Z(T) be its Hosoya index; let Fn be the nth Fibonacci number. It is proved in this work that Z(T)≤2F2mF2m+1 when T has order 4m with the equality holding if and only if T=T1,2m−1,2m−1, and that Z(T)≤F2m+22+F2mF2m+1 when T has order 4m+2 with the equality holding if and only if T=T1,2m+1,2m−1, where m is a positive integer and T1,s,t is a graph obtained by joining an isolated vertex with an edge to the (s+1)-th vertex (according to its natural ordering) of path Ps+t+1

An Algorithm for Computation of Bond Contributions of the Wiener Index

The Wiener index Wis usually obtained by adding distances between all pairs of vertices i and j (i, j = 1,2, ...N, where N denotes the total number of vertices). The Wiener index may also he obtained by adding all bond contributions We, i.e. W = L We, and We = L C\u27fj / Cij, where cij denotes the number of all the shortest paths between i and j that include edge e, and Cu denotes the total number of the shortest paths between vertices i and j. The summation has to be performed for all edges e and for all pairs of indices i and i. respectively. It is easy to calculate the bond contributions We for a cyclic graphs or for bridges. The present paper introduces an algorithm, which can be used to obtain bond contributions in cycle - containing graphs