Hosoya Index Research Articles

The smallest Hosoya index in (n, n + 1)-graphs

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. In this paper, we determine the lower bound for the Hosoya index in (n, n + 1)-graphs in terms of the order n, and characterize the (n, n + 1)-graph with the smallest Hosoya index.

On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter

The total number of matchings of a graph is defined as its Hosoya index. Conjugated and non-conjugated acyclic graphs that have maximal Hosoya index and short diameter are characterized in this paper, explicit expressions of the Hosoya indices of these extremal graphs are also presented.

Double hexagonal chains with maximal Hosoya index and minimal Merrifield–Simmons index

It is well known that the two graph invariants, “the Hosoya index” and “the Merrifield–Simmons index” are important ones in structural chemistry. The extremal hexagonal chains with respect to the Hosoya index and Merrifield–Simmons index are determined by Gutman and Zhang (J. Math. Chem., 12 (1993) 197–210, 27 (2000) 319–329 and J. Sys. Sci. Math. Sci., 18 (4) (1998) 460–465). In this paper, we will consider a type of the pericondensed hexagonal system. The double hexagonal chains with maximal Hosoya index and minimal Merrifield Simmons index are determined.

On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

Let G be an n-vertex unicyclic molecular graph and Z(G) be its Hosoya index, let F n be the nth Fibonacci number. It is proved in this paper that if G has girth l then Z(G) ≥ F l+1+(n−l)F l +F l-1, with the equality holding if and only if G is isomorphic to $$S_n^l$$ , the unicyclic graph obtained by pasting the unique non-1-valent vertex of the complete bipartite graph K 1,n-l to a vertex of an l-vertex cycle C l . A direct consequence of this observation is that the minimum Hosoya index of n-vertex unicyclic graphs is 2n−2 and the unique extremal unicyclic graph is $$S_n^3$$ . The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined.

The Merrifield–Simmons Indices and Hosoya Indices of Trees with k Pendant Vertices

The Merrifield–Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we characterize the trees with maximal Merrifield–Simmons indices and minimal Hosoya indices, respectively, among the trees with k pendant vertices.

Conjugated chemical trees with minimal energy and prescribed diameter

AbstractThe energy of a molecular graph G is defined as the sum of the absolute values of the eigenvalues of A(G), where A(G) is the adjacency matrix of this graph. This article characterizes conjugated chemical trees with prescribed diameter and minimal energies and presents explicit expressions of their Hosoya indices. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching

Let T be an acyclic graph without perfect matching and Z ( T ) be its Hosoya index; let F n be the n th Fibonacci number. It is proved in this work that Z ( T ) ≤ 2 F 2 m F 2 m + 1 when T has order 4 m with the equality holding if and only if T = T 1 , 2 m − 1 , 2 m − 1 , and that Z ( T ) ≤ F 2 m + 2 2 + F 2 m F 2 m + 1 when T has order 4 m + 2 with the equality holding if and only if T = T 1 , 2 m + 1 , 2 m − 1 , where m is a positive integer and T 1 , s , t is a graph obtained by joining an isolated vertex with an edge to the ( s + 1 ) -th vertex (according to its natural ordering) of path P s + t + 1 .

Wiener and Hosoya indices of reciprocal graphs

Two types of topological indices, Wiener and Hosoya indices, of three classes of reciprocal graphs (namely , and ) have been shown to be expressed in terms of the number of pendant vertices (n). The Wiener index is expressed in analytical form whereas the Hosoya index is expressed either by recurrence relations or directly by matrix products for which a facile computer program can be written. It has also been shown that the Hosoya index of can be obtained in analytical form.

On the minimal energy of trees with a given diameter

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. F. Zhang, H. Li [On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84] characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by I. Gutman [Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143]. In this letter, for a given positive integer d we characterize the tree with the minimal energy having diameter at least d . As a corollary, we also characterize the tree with the minimal Hosoya index having diameter at least d .

Coulson function and Hosoya index: extension of the relationship to polycyclic graphs and to new types of matching polynomials

Gutman et al. [Chem. Phys. Lett. 355 (2002) 378–382] established a relationship between the Coulson function, \(F(G,x) \,{=}\,n - [\hbox {i}x{\phi}^{\prime}(G,\hbox {i}x)/{\phi}(G,\hbox{i}x)]\), where \phi is the characteristic polynomial, and the Hosoya index Z, which is the sum over all k of the counts of all k-matchings. Like the original Coulson function, this relationship was postulated only for trees. The present study shows that the relationship can be extended to (poly)cyclic graphs by substituting the matching, or acyclic, polynomial for the characteristic polynomial. In addition, the relationship is extended to new types of matching polynomials that match cycles larger than edges (2-cyc1es). Finally, this presentation demonstrates a rigorous mathematical relationship between the graph adjacency matrices and the coefficients of these polynomials and describes computational algorithms for calculating them.

The higher‐order matching polynomial of a graph

Given a graph G with n vertices, let p(G, j) denote the number of ways j mutually nonincident edges can be selected in G. The polynomial , called the matching polynomial of G, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of length t, denoted by pt(G, j). We compare this higher‐order matching polynomial with the usual one, establishing similarities and differences. Some interesting examples are given. Finally, connections between our generalized matching polynomial and hypergeometric functions are found.

Extremal k*-cycle resonant hexagonal chains

Denote by ℬ*n the set of all k*-cycle resonant hexagonal chains with n hexagons. For any Bn∈ℬ*n, let m(Bn) and i(Bn) be the numbers of matchings (= the Hosoya index) and the number of independent sets (= the Merrifield–Simmons index) of Bn, respectively. In this paper, we give a characterization of the k*-cycle resonant hexagonal chains, and show that for any Bn∈ℬ*n, m(Hn)≤m(Bn) and i(Hn)≥i(Bn), where Hn is the helicene chain. Moreover, equalities hold only if Bn=Hn.

Extremal catacondensed benzenoids

Some results with respect to Hosoya index and Merrifield–Simmons index of tree-type hexagonal systems (catacondensed hydrocarbons) are shown. Using the results, the tree-type hexagonal systems with minimum, second minimum Hosoya index and maximum, second maximum Merrifield–Simmons index are determined. These results generalize some known results on extremal hexagonal chains.

Contemplation on the Hosoya indices

A vector model is presented based on the partial Hosoya indices. Two novel indices are introduced ( cos λ j and S j ). The model gives a better understanding of the relationship between the structure–activity related properties and the Hosoya indices.

Extremal Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

Coulson function and Hosoya index

Let G be a molecular graph, n the number of its vertices and φ( G, x) its characteristic polynomial. Already in 1940 Coulson expressed the total π-electron energy of conjugated unsaturated molecules in terms of the function F( x)= n−i xφ ′( G,i x)/ φ( G,i x). Recently, the Coulson function F( x) found applications also in modeling the structure-dependence of physico-chemical properties of alkanes. We now analyze the Coulson function and establish some of its hitherto unnoticed features, in particular its relations with the Hosoya index.

Extremal Chemical Trees

A variety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W. the largest graph eigenvalue λ1, the connectivity index X, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W , λ1, E, and Z. whereas the analogous problem for X was solved earlier. Among chemical trees with 5. 6, 7, and 3k + 2 vertices, k = 2,3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k +1 vertices, k = 3,4...., one tree has minimum 11 and maximum λ1 and another minimum E and Z .

On acyclic systems with minimal Hosoya index

The Hosoya index of a graph is defined as the total number of independent edge subsets of the graph. In this note, we characterize the trees with a given size of matching and having minimal and second minimal Hosoya index.

Extremal hexagonal chains concerning largest eigenvalue

In this paper, we define a roll-attaching operation of a hexagonal chain, and prove Gutman's conjecture affirmatively by using the operation. The idea of the proof is also applicable to the results concerning extremal hexagonal chains for the Hosoya index and Merrifield-Simmons index.

On interpretation of well-known topological indices.

Many topological indices lack an interpretation in terms of simple physicochemical quantities. We have reexamined the structural interpretation of well-known topological indices: the connectivity index (1)chi, the Wiener index W, and the Hosoya topological index Z. We relate the success of various indices in structure-property studies to the degree to which they differentiate contributions from more exposed terminal bonds and more buried interior bonds. When considering bond additive properties of alkanes we find better regressions when greater weights are assigned to terminal CC bonds and lesser weights to internal CC. We suggest here that topological indices be discussed in terms of their partitioning into bond contributions, which for different indices and different bonds will assume different values. With this insight we modified the Wiener index W into a new index W, in which bond contributions are determined using the reciprocal of the product of the number of atoms on each side of a bond. Similarly we modified the Hosoya index Z into Z in which the frequency of occurrence of individual CC bonds in the patterns of disjoint bonds are considered. Novel indices are compared with other indices and they show to yield better regressions for boiling points of octane isomers. This suggests a useful classification of topological indices based on the relative magnitudes of the contributions of terminal and interior bonds. To extend such considerations to other indices one needs to consider partitioning of global molecular indices into bond additive terms. A general scheme for partitioning of molecular descriptors into bond contributions is outlined for indices derived from a selection of matrices associated with molecular graphs.