It was recently shown that an interesting class of trees maximizes the MerrifieldSimmons index and minimizes the Hosoya index and energy among all trees with given number of vertices and maximum degree. In this paper, we describe how these trees (which we will call F-trees) can be constructed algorithmically by means of so-called F-expansions, which are very similar to ordinary base-d digital e...

Abstract Many chemical indices have been invented in theoretical chemistry, such as Wiener index, Merrifield-Simmons index, Hosoya index, spectral radius and Randić index, etc. The extremal trees and unicyclic graphs for these chemical indices are interested in existing literature. Let G be a molecular graph (called a cacti), which all of blocks of G are either edges or cycles. Denote G (n, r) ...

The sum of distances between all vertices pairs in a connected graph is known as the Wiener Index. It is the earliest of the indices that correlates well with many physicochemical 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, e...

let $g$ be a molecular graph with vertex set $v(g)$, $d_g(u, v)$ the topological distance between vertices $u$ and $v$ in $g$. the hosoya polynomial $h(g, x)$ of $g$ is a polynomial $sumlimits_{{u, v}subseteq v(g)}x^{d_g(u, v)}$ in variable $x$. in this paper, we obtain an explicit analytical expression for the expected value of the hosoya polynomial of a random benzenoid chain with $n$ hexagon...

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di σi denote, respectively, degree transmission of vi, for 1≤i≤n. In this paper, we aim to provide new matrix description celebrated Wiener index. fact, introduce Wiener–Hosoya G, which is defined as n×n whose (i,j)-entry equal σi2di+σj2dj if vi vj are adjacent 0 otherwise. Some properties, including upper lower ...

Graphical matrices are presented. Their construction via selected sets of subgraphs and the replacement of subgraphs by numbers representing graph invariants are discussed. The last step of the procedure is to apply the method of choice for obtaining the desired double invariant from the graphical matrix in the numerical form. It is also pointed out that many so-called special graph-theoretical...