The Wiener polynomial of a graph

The Wiener Polynomial of a Graph

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...

Peripheral Wiener Index of a Graph

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...

The Hyper-Wiener Polynomial of Graphs

The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph $G$ is equal to the length of a shortest path that connects $u$ and $v$. Define $WW(G,x) = 1/2sum_{{ a,b } subseteq V(G)}x^{d(a,b) + d^2(a,b)}$, where $d(G)$ is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are compu...

On the Roots of Hosoya Polynomial of a Graph

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

The Wiener Polynomial of the kth Power Graph