Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph

The determination of Hosoya polynomial is the latest scheme, and it provides an excellent and superior role in finding the Weiner and hyper-Wiener index. The application of Weiner index ranges from the introduction of the concept of information theoretic analogues of topological indices to the use as major tool in crystal and polymer studies. In this paper, we will compute the Hosoya polynomial for multiwheel graph and uniform subdivision of multiwheel graph. Furthermore, we will derive two well-known topological indices for the abovementioned graphs, first Weiner index, and second hyper-Wiener index.

Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y : x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.

Hosoya properties of the commuting graph associated with the group of symmetries

A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index or Z-index of a graph G is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).

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.

Hosoya and Harary Polynomials of Hourglass and Rhombic Benzenoid Systems

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Harry Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we compute the Hosoya polynomials for hourglass and rhombic benzenoid systems and recover Wiener and hyper-Wiener indices from them.

Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.