Hosoya Index Research Articles

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.

On the Hosoya index of graphs

Let G be a (molecular) graph. The Hosoya index Z(G) of G is defined as the number of subsets of the edge set E(G) in which no two edges are adjacent in G, i.e., Z(G) is the total number of matchings of G. In this paper, we determine all the connected graphs G with n + 1 ≤ Z(G) ≤ 5n − 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.

On the Hosoya index and the Merrifield–Simmons index of graphs with a given clique number

The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let W n , k be the set of connected graphs with n vertices and clique number k . In this note we characterize the graphs from W n , k with extremal (maximal and minimal) Hosoya indices and the ones with extremal (maximal and minimal) Merrifield–Simmons indices, respectively.

ON MAXIMAL ENERGY AND HOSOYA INDEX OF TREES WITHOUT PERFECT MATCHING

AbstractLet G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

Vertex topological indices and tree expressions, generalizations of continued fractions

We expand on the work of Hosoya to describe a generalization of continued fractions called "tree expressions." Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a "vertex topological index" which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to "labeled tree expressions" that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.

On the extremal Merrifield–Simmons index and Hosoya index of quasi-tree graphs

It is well known that the two graph invariants, “the Merrifield–Simmons index” and “the Hosoya index” are important in structural chemistry. A graph G is called a quasi-tree graph, if there exists u 0 in V ( G ) such that G − u 0 is a tree. In this paper, at first we characterize the n -vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield–Simmons indices. Then we characterize the n -vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those n -vertex quasi-tree graphs with k pendent vertices having the smallest Hosoya index.

Ordering trees with given pendent vertices with respect to Merrifield-Simmons indices and Hosoya indices

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 order a kind of trees with given number of pendent vertices with respect to Merrifield-Simmons indices and Hosoya indices.

Chemical trees minimizing energy and Hosoya index

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.

The largest Hosoya index of (n,n+1)-graphs

The Hosoya index of a graph is defined as the total number of its matchings. In this paper, we obtain that the largest Hosoya index of ( n , n + 1 ) -graphs is f ( n + 1 ) + f ( n − 1 ) + 2 f ( n − 3 ) , where f ( n ) is the n th Fibonacci number, and we characterize the extremal graphs.

On extremal unicyclic molecular graphs with maximal Hosoya index

Let G be a unicyclic n -vertex graph and Z ( G ) be its Hosoya index, let F n stand for the n th Fibonacci number. It is proved in this paper that Z ( G ) ≤ F n + 1 + F n − 1 with the equality holding if and only if G is isomorphic to C n , the n -vertex cycle, and that if G ≠ C n then Z ( G ) ≤ F n + 1 + 2 F n − 3 with the equality holding if and only if G = Q n or D n , where graph Q n is obtained by pasting one endpoint of a 3-vertex path to a vertex of C n − 2 and D n is obtained by pasting one endpoint of an ( n − 3 ) -vertex path to a vertex of C 4 .

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S ( G ) be the subdivision graph of G , which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S ( G ) in terms of the matchings of G . Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S ( G ) are obtained. As an application, we prove a combinatorial identity.

The Merrifield-Simmons indices and Hosoya indices of some classes of cartesian graph product

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 give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K2 ◊ H, where H is a path graph Pn, cyclic graph Cn, or star graph Sn, with n vertices (These are called: ladder graph, prism graph, and book graph).

Extremal Hosoya index and Merrifield–Simmons index of hexagonal spiders

For any graph G, let m(G) and i(G) be the numbers of matchings (i.e., the Hosoya index) and the number of independent sets (i.e., the Merrifield–Simmons index) of G, respectively. In this paper, we show that the linear hexagonal spider and zig-zag hexagonal spider attain the extremal values of Hosoya index and Merrifield–Simmons index, respectively.

Trees with [formula omitted]-matchings and the fourth and fifth minimal Hosoya index

We study the Hosoya index of trees with m -matchings and characterize the trees with m -matchings and the fourth and fifth minimal Hosoya index respectively.

Bounds of the Hosoya index in graphs

A subset M of the edge set E (G) of a graph G is a matching if no two edges of M are adjacent in G. In this paper we study the number of matchings in graphs, that is the Hosoya index. We study the problem of maximizing and minimizing the Hosoya index in a graph. For specific graphs this topological index is maximized by the Fibonacci number or the Lucas number. Next we give the generalizations of the Fibonacci number and the Lucas number which give the number of matchings in special multigraphs.

Minimizing a class of unicyclic graphs by means of Hosoya index

The Hosoya index, denoted by z ( G ) , of a (molecular) graph G is defined as the total number of independent-edge sets of G . Let U n be the set of unicyclic graphs with n vertices. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. Denote by U n 1 the set of fully loaded unicyclic graphs. In this paper, graphs in U n 1 with minimal, second-minimal and third-minimal Hosoya indices are uniquely determined, respectively.

The proof of a conjecture concerning acyclic molecular graphs with maximal Hosoya index and diameter 4

The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we solve a conjecture in Ou, J. Math. Chem, DOI: 10.1007/S10910-006-9199-1 concerning acyclic molecular graphs with maximal Hosoya index and diameter 4.

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let $$S_{n}^{l}$$ be the graph obtained by identifying the center of the star S n-l+1 with any vertex of C l . By $$R_{n}^{l,\,k}$$ we denote the graph obtained by identifying one pendent vertex of the path P n-l-k+1 with one pendent vertex of $$S_{l+k}^{l}$$ . In this paper, we show that $$R_{n}^{l,\,k}$$ is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).

Extremal double hexagonal chains with respect to k-matchings and k-independent sets

“Double hexagonal chains” can be considered as benzenoids constructed by successive fusions of successive naphthalenes along a zig-zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. In this paper, we discuss the numbers of k-matchings and k-independent sets of double hexagonal chains, as well as Hosoya indices and Merrifield–Simmons indices, and obtain some extremal results: among all the double hexagonal chains with the same number of naphthalene units, (a) the double linear hexagonal chain has minimal k-matching number and maximal k-independent set number and (b) the double zig-zag hexagonal chain has maximal k-matching number and minimal k-independent set number, which are extensions to hexagonal chains [L. Zhang and F. Zhang, Extremal hexagonal chains concerning k-matchings and k-independent sets, J. Math. Chem. 27 (2000) 319–329].

Important Mathematical Structures of the Topological Index Z for Tree Graphs

Mathematical importance of the topological index, ZG, or the so-called Hosoya index is stressed by presenting and giving supporting evidence for the proposed conjecture. That is, for a given pair of positive integers (n1<n2) which are prime with each other there exists a series of Z-trees {Gm} of the property, Z(Gm) = Z(G(m-1)) + Z(G(m-2)) (m>or=3), with Z(G1) = n1 and Z(G2) = n2.