Let G be a graph with p vertices and q edges and A = {0, 1, 2, . . . , [q/2]}. A vertex labeling f : V (G) → A induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the indu...

The Wiener index of a connected graph is the sum distances between all unordered pairs vertices. A Eulerian if its vertex degrees are even. In Gutman et al. (2014) authors proved that cycle unique maximising among graphs given order. They also conjectured for order n≥26 consisting on n−2 vertices and triangle share with second largest index. conjecture known to hold n≤25 exception six values. t...

Let G be a connected graph and ξ(G) = Sze(G)−We(G), where We(G) denotes the edge Wiener index and Sze(G) denotes the edge Szeged index of G. In an earlier paper, it is proved that if T is a tree then Sze(T ) = We(T ). In this paper, we continue our work to prove that for every connected graph G, Sze(G) ≥ We(G) with equality if and only if G is a tree. We also classify all graphs with ξ(G) ≤ 5. ...

For a graph G, let σ(G) = ∑ uv∈E(G) 1 √ dG(u)+dG(v) ; this defines the sum-connectivity index σ(G). More generally, given a positive function t, the edge-weight t-index t(G) is given by t(G) = ∑ uv∈E(G) t(ω(uv)), where ω(uv) = dG(u) + dG(v). We consider extremal problems for the t-index over various families of graphs. The sum-connectivity index satisfies the conditions imposed on t in each ext...

One of the generalizations of the Wiener number to weighted graphs is to assign probabilities to edges, meaning that in nonstatic conditions the edge is present only with some probability. The Reliability Wiener number is defined as the sum of reliabilities among pairs of vertices, where the reliability of a pair is the reliability of the most reliable path. Closed expressions are derived for t...

abstract. suppose g is an nvertex and medge simple graph with edge set e(g). an integervalued function f: e(g) → z is called a flow. tutte was introduced the flow polynomial f(g, λ) as a polynomial in an indeterminate λ with integer coefficients by f(g,λ) in this paper the flow polynomial of some dendrimers are computed.

The edge and total versions of geometric-arithmetic (GA) index of graphs are introduced based on the end-vertex degrees of edges of their line and total graphs, respectively. In this paper, the edge and total GA indices are computed for some graphs by using some results on GA index and graphs.

A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for ...

Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, (edge-)Mostar and (vertex-)PI index. For these indices, corresponding polynomials were also defined, i.e., polynomial, Mostar PI etc. It is well known that, by evaluating first derivative of such a polynomial at x=1, we obtain related The aim this paper to ...

fullerenes are closed−cage carbon molecules formed by 12 pentagonal and n/2 – 10hexagonal faces, where n is the number of carbon atoms. patrick fowler in his lecture inmcc 2009 asked about the wiener index of fullerenes in general. in this paper werespond partially to this question for an infinite class of fullerenes with exactly 10ncarbon atoms. our method is general and can be applied to full...