The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q-analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial, extends this concept by trying to capture the complete distribution of distances in the graph.Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.

Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined as the product of weights of vertices and edges of $T_1$. We obtain a linear-time algorithm to count the sum of weights of subtrees of $T$. As applications, we characterize the tree with the diameter at least $d$, which has the maximum number of subtrees, and we characterize the tree with the maximum degree at least $\Delta$, which has the minimum number of subtrees.Comment: 20 pages, 11 figure

The energy of a simple graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of eigenvalues of G. As the dimer problem and spanning trees problem in statistical physics, in this paper we propose the energy per vertex problem for lattice systems. In general for a type of lattices in statistical physics, to compute the entropy constant with toroidal, cylindrical, Mobius-band, Kleinbottle, and free boundary conditions are different tasks with different hardness and may have different solution. We show that the energy per vertex of plane lattices is independent on the toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions. Particularly, the asymptotic formulae of energies of the triangular, 3 3 .4 2 , and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are obtained explicitly. , 15]), the spanning tree problem considers the entropy of spanning trees [2,14,17], and the independent set problem considers the entropy of independent sets [1]. It is natural to consider the chemical physics parameter -

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