The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4], [5]) states that for any positive integer n (except numbers from a given 49 element set), one can find a tree with Wiener index n. In this paper, we prove that every i...

For a connected graph G and an non-empty set S ⊆ V (G), the Steiner distance dG(S) among the vertices of S is defined as the minimum size among all connected subgraphs whose vertex sets contain S. This concept represents a natural generalization of the concept of classical graph distance. Recently, the Steiner Wiener index of a graph was introduced by replacing the classical graph distance used...

The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. In [1], the tree that minimizes the Wiener index among trees of given maximal degree is studied. We characterize trees that achieve the maxim...

The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. In [1], the tree that minimizes the Wiener index among trees of given maximal degree is studied. We characterize trees that achieve the maxim...

the edge szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph g, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. in this paper, the edge szeged index of one-pentagonal carbon nanocone cnc5[n] is computed for the first ...

The eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs. In t...

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperti...

in this paper, a novel topological index, named m-index, is introduced based on expanded form of the wiener matrix. for constructing this index the atomic characteristics and the interaction of the vertices in a molecule are taken into account. the usefulness of the m-index is demonstrated by several qspr/qsar models for different physico-chemical properties and biological activities of a large...