The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing ...multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.
Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are ...related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized.
The Hosoya index and the Merrifield‐Simmons index are two important molecular descriptors in chemical graph theory. The Hosoya index is defined as the total number of matchings of the graph and the ...Merrifield‐Simmons index is defined as the total number of independent sets of the graph. In this paper, the author obtains the extremal values of the Hosoya index and the Merrifield‐Simmons index of trees with given domination number.
In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental ...polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered rings are identified.
Hosoya index of tree structures Ramin Kazemi; Ali Behtoei
Transactions on combinatorics,
09/2020, Letnik:
9, Številka:
3
Journal Article
Recenzirano
Odprti dostop
The Hosoya index, also known as the $Z$ index, of a graph is the total number of matchings in it. In this paper, we study the Hosoya index of the tree structures. Our aim is to give ...some results on $Z$ in terms of Fibonacci numbers in such structures. Also, the asymptotic normality of this index is given.
In this paper, we calculate the Hosoya index in a family of deterministic recursive trees with a special feature that includes new nodes which are connected to existing nodes with a certain rule. We ...then obtain a recursive solution of the Hosoya index based on the operations of a determinant. The computational complexity of our proposed algorithm is O(log2n) with n being the network size, which is lower than that of the existing numerical methods. Finally, we give a weighted tree shrinking method as a graphical interpretation of the recurrence formula for the Hosoya index.
•Calculating the Hosoya index of a family of deterministic recursive trees.•Obtaining a solution of the Hosoya index based on the operations of a determinant.•The computational complexity of our proposed algorithm is lower than that of the existing numerical methods.
Suppose G is a finite group. The power graph represented by P(G) of G is a graph, whose node set is G, and two different elements are adjacent if and only if one is an integral power of the other. ...The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph Γ is known as the
-index or Hosoya index. The
-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.
Hosoya introduced the concept of graph terminologies in chemistry and provide a modeling for molecules. This modeling leads to predict the chemical properties of molecules, easy classification of ...chemical compounds, computer simulations and computer-assisted design of new chemical compounds. In this article, we determine the non-commuting graph associated with the dihedral group by using three Hosoya parameters (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index). These Hosoya parameters contain a pile of information about distance structure as well as the edge independent structure of the above-mentioned graph.
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