Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of P, and Ept(P) is the EPT graph (i.e. the edge intersection graph) of the paths P of the tree T . These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least...

The edge-Wiener index of a connected graph is the sum of the distances between all pairs of edges of the graph. In this paper, we determine the polyphenyl chains with minimum and maximum edge-Wiener indices among all the polyphenyl chains with h hexagons. Moreover, explicit formulas for the edge-Wiener indices of extremal polyphenyl chains are obtained.

We study the following problem: Given a set V of n vertices and a set E of m edge pairs, we define a graph family G(V, E) as the set of graphs that have vertex set V and contain exactly one edge from every pair in E . We want to find a graph in G(V, E) that has the minimal number of connected components. We show that, if the edge pairs in E are non-disjoint, the problem is NP-hard. This is true...

Let G be a simple connected graph in chemical graph theory and uν e = be an edge of G. The Randić index ( ) G χ and sum-connectivity ( ) G X of a nontrivial connected graph G are defined as the sum of the weights ν ud d 1 and ν u d d + 1 over all edges ν u e = of G, respectively. In this paper, we compute Randić ( ) G χ and sum-connectivity ( ) G X indices of V-phenylenic nanotubes and nanotori.

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+ d(v)+d(w)~> n + 2 for every triple u, v, w of independent vertices. As a corollary we obtain the following improvement of a conjecture of H/iggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and ass...

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper...

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper...

In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is ...

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper...

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Two graphs of the same order are said to be equienergetic if their energies are equal. We point out the following two open problems for equienergetic graphs. (1) Although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...