The Reciprocal Complementary Wiener Number of Graph Operations

The reciprocal complementary Wiener number of a connected graph G is defined as ∑ {x,y}⊆V (G) 1 D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

Hosoya Polynomial for Subdivided Caterpillar Graphs

Background: Computing Hosoya polynomial for the graph associated with the chemical compound plays a vital role in the field of chemistry. From Hosoya polynomial, it is easy to compute Weiner index(Weiner number) and Hyper Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected by the physicochemical properties of chemical compounds. Caterpillar trees have been used in chemical graph theory to represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices. Method: The Hosoya polynomial of a graph G is defined as H(G;x)=∑_(k=0)d(G) d(G,k) x^k . In order to compute the Hosoya polynomial, we need to find its coefficients d(G,k) which is the number of pair of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance m,2≤m≤(n+1)k in the form of sets. Then finding the cardinality of these sets and adding up will give us the value of coefficient d(G,m). Finally using the values of coefficients in the definition we get the Hosoya polynomial of Uniform subdivision of caterpillar graph. Result: In this work we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph. Conclusion: Caterpillar trees are among one of the important and general classes of trees. Thorn rods and thorn stars are the important subclasses of caterpillar trees. The ideas of the present research article is to give a road map to those researchers who are interesting to study the Hosoya polynomial for different trees.

Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems

In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge LC of these nano-structures, which have a constant length LM of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, LC, LM) over a certain threshold size LC. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size LC ≥ LC, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.

The reciprocal complementary Wiener number of graphs

The reciprocal complementary Wiener number (RCW) of a connected graph G is defined as the sum ofweights frac{1}{D+1-d_G(x,y)} over all unordered vertex pairs in a graph G, where D is the diameter of Gand d_G(x,y) is the distance between vertices x and y. In this paper, we find new bounds for RCW ofgraphs, and study this invariant of two important types of graphs, named the Bar-Polyhex and theMycielskian graphs.

Bounds on hyper-Wiener index of graphs

The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper and lower bounds for [Formula: see text], in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.