The Wiener index is a graph invariant that has found extensive application in chemistry. In addition to that a generating function, which was called the Wiener polynomial, who’s derivate is a q-analog of the Wiener index was defined. In an article, Sagan, Yeh and Zhang in [The Wiener Polynomial of a graph, Int. J. Quantun Chem., 60 (1996), 959969] attained what graph operations do to the Wiene...

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP2) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algo...

If P is a given graph property, we say that a graph G is locally P if 〈N(v)〉 has property P for every v ∈ V (G) where 〈N(v)〉 is the induced graph on the open neighbourhood of the vertex v. Pareek and Skupień (C. M. Pareek and Z. Skupień, On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 17, 1983) posed the following two questions. Question 1 Is 9 the smalle...

Let G = (V,E) be a simple undirected graph. N(G) = (V,EN ) is the neighborhood graph of G, if and only if EN = {{a, b} ∣ a ∕= b ∧ ∃x ∈ V : {x, a} ∈ E ∧ {x, b} ∈ E}. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph Nk(G) := N(N(. . . N(G))) of G. So we det...

A connected k-chromatic graph G is double-critical if for all edges uv of G the graph G−u−v is (k−2)-colourable. The only known double-critical k-chromatic graph is the complete k-graph Kk. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erd®s and Lovász. The conjecture has been veri ed for k at most 5. We prove for k = 6 and k =...

We introduce the notion of uniform number of a graph. The  uniform number of a connected graph $G$ is the least cardinality of a nonempty subset $M$ of the vertex set of $G$ for which the function $f_M: M^crightarrow mathcal{P}(X) - {emptyset}$ defined as $f_M(x) = {D(x, y): y in M}$ is a constant function, where $D(x, y)$ is the detour distance between $x$ and $y$ in $G$ and $mathcal{P}(X)$ ...

A connected k-chromatic graph G is double-critical if for all edges uv of G the graph G − u − v is (k − 2)-colourable. The only known double-critical k-chromatic graph is the complete k-graph Kk. The conjecture that there are no other doublecritical graphs is a special case of a conjecture from 1966, due to Erdős and Lovász. The conjecture has been verified for k at most 5. We prove for k = 6 a...

in this paper an algorithm for computing the balaban and randic indices of any simple connected graph was introduced. also these indices were computed for ipr c80 fullerene isomers, zigzag nanotubes and graphene by gap program.

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...