On the PI polynomial of a graph

Applied Mathematics Letters, 2007

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.

International Journal of Quantum Chemistry, 1996

The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincaré polynomial of a finite Coxeter group.

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 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. The mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. Herein we show how the Wiener Index of a graph changes with these operations, and extend the results to Wiener Polynomials.

2017

Applied and Computational Mathematics, 2016

Let G be a simple connected graph having vertex set V and edge set E. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The length of the smallest path between vertices u,v∈ V(G) is called the distance, d(u,v), between the vertices u,v. Mathematical chemistry is the area of research engaged in new application of mathematics in chemistry. In mathematics chemistry, we have many topological indices for any molecular graph, that they are invariant on the graph automorphism. In this research paper, we computing the Wiener index and the Hosoya polynomial of the Jahangir graphs J 5,m for all integer number m≥3. The Wiener index is the sum of distances between all pairs of vertices of G as W(G)=

2017

Let G = (V,E) be a graph. The total graph T (G) of G is that graph whose vertex set is V ∪ E, and two vertices are adjacent if and only if they are adjacent or incident in G. For a graph G = (V,E), the graph G.Sm is obtained by identifying each vertex of G by a root vertex of Sm and the graph Sm.G is obtained by identifying each vertex of Sm except root vertex by any vertex of G, where Sm is a star graph with m vertices. In this paper, we consider G as the cycle graph Cn with n vertices and investigate the Wiener index of the total graphs of Cn.Sm and Sn.Cm. MSC: 05C12, 05C76

Let us consider G be a graph and e = uv an edge of G. Denote by nu(e) the number of vertices lying closer to the vertex u than the vertex v, and define n v (e) by analogy. By definition, the vertex PI index of G is given by P I v (G) = X e=uv∈E(G) [n u (e) + n v (e)]. Suppose G is a triangle free graph. In this paper, it is proved that P I v (G) ≥ M 1 (G) with equality if and only if diam(G) = 2, where M 1 (G) = P v∈V (G) deg G (v) 2 is the first Zagreb index and diam(G) denotes the diameter of the graph G. Moreover, we show that the condition of "triangle free" cannot be omitted.

Proceedings of the 2nd Croatian Combinatorial Days, 2019

The Wiener index W (G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991,Šoltés [9] posed the problem of finding all graphs G such that equality W (G) = W (G − v) holds for all vertices v in G. The only known graph with this property is the cycle C 11. Our main object of study is the relaxed version of this problem: find graphs for which Wiener index does not change when a particular vertex v is removed. This overview contains results which were obtained and published during the past two years concerning relaxedŠoltés's problem.

Informatics Engineering, an International Journal

Let G be a graph. The distance d(u,v)

Advances and Applications in Discrete Mathematics, 2019

Let G be a connected graph. The Wiener index of a graph is defined as the sum of all distances between different vertices, and the Hosoya polynomial of a graph G is defined as (,) { , } () (,)