## Wiener index of path graph

Wiener index. The Wiener index of a vertex is the sum of the shortest path distances between v and all other vertices. The Wiener index of a graph G is the sum of the shortest path distances over all pairs of vertices. Used by mathematical chemists (vertices = atoms, edges = bonds). Random walk. networkx.algorithms.wiener.wiener_index¶ wiener_index (G, weight=None) [source] ¶. Returns the Wiener index of the given graph. The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.

The distance of between any two vertices u and v of graph is defined as the length of the shortest path connecting u, v is d (u, v).A topological index is a number. Wiener Index of a Cycle in the Context of Some Graph Operations. 3. • Graphs with boxicity k ( S is the set of boxes of dimension k ). • Line graphs ( S is the set of  Keywords: Degree Distance; Wiener Index; Cacti. 1. The Wiener index of a graph is the sum of the neighbor of x on the shortest path between x and y and. 3 Oct 2015 Among 2-connected graphs on n vertices (or even stronger, among the graphs of minimum degree 2), the n-cycle has the largest Wiener index. 1 Jan 1995 Three Methods for Calculation of the Hyper-Wiener Index of Molecular Graphs. Journal of Chemical Information and Computer Sciences 2002,  13 May 2019 Peripheral Wiener index of a graph is the sum of the distance of v), if there is no ambiguity), which is defined as the length of a shortest path. [5, 7]. Among connected graphs of order n, the path Pn has maximum Wiener index . Therefore in the class of 1-connected and 1-edge–connected graphs.

## where dG.u;v/ is the shortest path distance between vertices u and v in G. Details on the Wiener index can be found in [7, 8, 12, 14, 15]. Edge versions of the

22 Jun 2017 Keywords: Distance in graphs, Wiener index, peripheral Wiener index two vertices u and v of G is the length of a shortest path between u and  10 Jul 2019 graph. In chemical graph theory the Wiener index of a graph G, denoted by W(G) and Hyper Wiener indices of Unitary addition Cayley graph Gn is computed. of hyper-Wiener index for cycle-containing structures, J. Chem. 13 Jul 2016 The Wiener index is the sum of distances between all pairs of m≥3; i.e., a graph consisting of a cycle Cnm with one additional vertex which is  15 Jan 2015 Hyper-Wiener index of gear fan graph, gear wheel graph and their r-corona vertex in every two adjacent vertices of the fan path Pn of fan. of the graph G and ij d is the distance (i.e., the number of edges of a shortest path ) between the vertices i v and j v . The relation between Wiener index. ( ). The Wiener index for complete graph is. 2. 2. n n. -. , and Wiener index for path graph is. 3. 6 n n. -. , The wiener index for the cycle graph is w(Cn)=. 2−1.

### various applications in physical chemistry [7, 9]. Let F1be the 5-vertex path, F2the graph obtained by identifying a vertex.

10 Nov 2015 The Wiener index of molecular graph G was defined as. (1) We set P n, C n, and K 1,n−1 as the path, cycle, and the star with order (vertex  The total distance or Wiener index W(G) of a connected graph G is defined as of edges on a shortest path connecting these vertices in G, and the distance,. 1 Sep 2008 The distance d(u,v) between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v. The Wiener index  22 Jun 2017 Keywords: Distance in graphs, Wiener index, peripheral Wiener index two vertices u and v of G is the length of a shortest path between u and

### The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius

where dG.u;v/ is the shortest path distance between vertices u and v in G. Details on the Wiener index can be found in [7, 8, 12, 14, 15]. Edge versions of the  The distance of between any two vertices u and v of graph is defined as the length of the shortest path connecting u, v is d (u, v).A topological index is a number. Wiener Index of a Cycle in the Context of Some Graph Operations. 3. • Graphs with boxicity k ( S is the set of boxes of dimension k ). • Line graphs ( S is the set of  Keywords: Degree Distance; Wiener Index; Cacti. 1. The Wiener index of a graph is the sum of the neighbor of x on the shortest path between x and y and. 3 Oct 2015 Among 2-connected graphs on n vertices (or even stronger, among the graphs of minimum degree 2), the n-cycle has the largest Wiener index. 1 Jan 1995 Three Methods for Calculation of the Hyper-Wiener Index of Molecular Graphs. Journal of Chemical Information and Computer Sciences 2002,

## We show that among all graphs on n vertices which have p≥2 blocks, the maximum Wiener index is attained by a graph composed of two cycles joined by a path (here we admit that one or both cycles

networkx.algorithms.wiener.wiener_index¶ wiener_index (G, weight=None) [source] ¶. Returns the Wiener index of the given graph. The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted. The Wiener index W(G) of a connected graph G is the sum of the distances between all pairs (ordered) of vertices of G. ∑, , . In this paper, we give theoretical results for calculating the Wiener index of a cycle in the context of some graph operations. These formulas will pave the way to demonstrate the Wiener index of molecular structures. The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius The Wiener index of a graph G is defined as W(G)=∑ u,v d G (u,v), where d G (u,v) is the distance between u and v in G, and the sum goes over all pairs of vertices.

The Wiener index , denoted (Wiener 1947) and also known as the "path number" or Wiener number (Plavšić et al. 1993), is a graph index defined for a graph on nodes by where is the graph distance matrix . Wiener index of a graph 1.1 Introduction Let G = (V(G),E(G)) be a simple connected undirected graph. For sub-sequent discussions we will always consider such graphs only. The Wiener index or Wiener number W(G) of G is deﬁned as W (G)= 1 2 u∈V (G) v∈V (G) d G (u,v). (1.1) Here, d G (u,v) (or simply d(u,v) when no confusion arises) denotes the Wiener index of trees As the path and therefore the distance between two vertices of a tree is unique, the Wiener index of a tree is much easier to compute than that of an arbitrary graph. On the Wiener index of a graph. Abstract. A modification of the Weiner index which properly takes into account the symmetry of a graph is proposed. The explicit formulae for the modified Wiener index of path, cycle, complete bipartite, cube and lattice graphs are derived and compared with their standard Wiener index. The Wiener index of the graph G, denoted by W = W (G), is the sum of distances between all pairs of vertices of G. The Wiener index of graphs has been studied in much detail (see the reviews , , , the recent papers , , , , , , , , and the references cited therein). Yet, Wiener indices of Eulerian graphs seem to have evaded the attention of scholars. “The Wiener index of a graph is represented by and defined as the sum of distances between all pairs of vertices in a simple graph ”: Based on the Wiener index, Hosoya introduced the Wiener polynomial (now called Hosoya polynomial) in 1988 [ 8 ].