Merrifield-Simmons Index Research Articles

Cacti with the maximum Merrifield–Simmons index and given number of cut edges

The Merrifield–Simmons index of a graph G, denoted by i(G), is defined to be the total number of its independent sets, including the empty set. A graph G is called a cactus if each block of G is either an edge or a cycle. Denote by C(n,k) the set of connected cacti possessing n vertices and k cut edges. In this work, we shall characterize the cacti with the maximum Merrifield–Simmons index among all graphs in C(n,k) for all possible values of k.

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 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.

Tricyclic graphs with maximum Merrifield–Simmons index

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for the Merrifield–Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246–256]. The sharp upper bound for the Merrifield–Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield–Simmons index in ( n , n + 1 ) -graphs, J. Math. Chem. 43 (1) (2008) 75–91]. The sharp lower bound for the Merrifield–Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield–Simmons index is determined.

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 smallest Merrifield–Simmons index of [formula omitted]-graphs

The Merrifield–Simmons index of a graph is defined as the total number of its independent sets. In this paper, we show that the smallest Merrifield–Simmons index of ( n , n + 1 ) -graphs is f ( n ) + 2 f ( n − 2 ) + f ( n − 4 ) , where f ( n ) is the n th Fibonacci number, and characterize the extremal graphs.

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.

Unicycle graphs with extremal Merrifield–Simmons Index

Let G = (V(G),E(G)) denote a graph whose set of vertices and set of edges are V(G)and E(G), respectively. For any v ∈ V(G), we denote the neighbors of v as N(v) .B yn(G), we denote the number of vertices of G. All graphs considered here are both finite and simple. We denote, respectively, by Sn, Pn, and Cn the star, path, and cycle with n vertices. Let (G1,v1) and (G2,v2) be two graphs rooted at v1 and v2, respectively, then G = (G1,v1) �� (G1,v2) denote the graph obtained by identifying v1 with v2 as one common vertex. Let Un denote the set of all unicycle graphs of order n .B yU(n,k) we denote the set of unicycle graphs in which the length of its cycle is k. For any graph G in U(n,k), we denote the unique cycle of length k in G as Ck. Other notations and terminology not defined here will conform to those in [1]. For any given graph G, its Merrifield‐Simmons index, simply denoted as f(G), is defined to be the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., in graph-theoretical terminology, the number of independent-vertex subsets of G, including the empty set. We take the cycle C4 =

The Merrifield–Simmons 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 upper bound for the Merrifield–Simmons index in (n, n + 1)−graphs in terms of the order n, and characterize the (n, n+1)−graph with the largest Merrifield–Simmons index.

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.

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.

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.

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.

Extremal hexagonal chains

Some extremal properties of the linear chainLh ofh hexagons are pointed out. In the class of all hexagonal chains withh hexagons,Lh has minimumK,Z andx1 values, as well as maximum W and σ values;K = number of perfect matchings,Z = number of independent edge sets (Hosoya index),x1 = largest graph eigenvalue,W = Wiener index, σ= number of independent vertex sets (Merrifield-Simmons index). The extremality ofLh with respect toZ, σ andx1 is demonstrated here for the first time.