Let T be an acyclic molecule with n vertices, and let S(T ) be the acyclic molecule obtained from T by replacing each edge of T by a path of length two. In this work, we show that the Wiener index of T can be explained as the number of matchings with n− 2 edges in S(T ). Furthermore, some related results are also obtained.

A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20.

The synthesis and anion binding properties of novel halogen-bonding (XB) bis-iodotriazole-pyridinium-containing acyclic and [2]catenane anion host systems are described. The XB acyclic receptor displays selectivity for acetate over halides with enhanced anion recognition properties compared to the analogous hydrogen-bonding (HB) acyclic receptor. A reversal in halide selectivity is observed in ...

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G and is denoted by η(G) or η. Also, If every me...

In this report we obtain (good) upper bounds on the acyclic chromatic index of graphs obtained as the tensor product of two (or more) graphs, in terms of the acyclic chromatic indices of those factor graphs in the tensor product. Our results assume that optimal colourings for the factor graphs are given and our algorithms extend these colourings to obtain good (close to optimal) colourings of t...

In this paper, we investigate some properties on capacity factors, which were proposed to investigate the link failure problem from network coding. A capacity factor (CF) of a network is an edge set, deleting which will cause the maximum flow to decrease while deleting any proper subset will not. Generally, a k-CF is a minimal (not minimum) edge set which will cause the network maximum flow dec...

Consider a weighted n-vertex, m-edge graph G with designated source s and destination t. The topology of G is known, while the edge weights are hidden. Our goal is to discover either the edge weights in the graph or a shortest (s, t)-path. This is done by means of agents that traverse different (s, t)-paths in multiple rounds and report back the total cost they incurred. Various cost models are...

Given an embedded planar acyclic digraph G, the acyclic hamiltonian path completion with crossing minimization (AcyclicHPCCM) problem is to determine a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. In this paper, we present a linear time algorithm whi...

For overcoming the problems such as distortion and shift of object’s edge, easily losing the object detail information of methods in image edge detection, and satisfying higher demand for object detection in the modern war, a new image edge detection method was designed. LOG edge detection as a typical image processing method was introduced and the disadvantage of this model was analyzed firstl...