The complexity of the Clar number problem and an exact algorithm

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.     

Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C _l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for and l = 2r + 1 or has the minimal energy in the graphs exclusive of , where is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C ₄ and then by attaching n/2 − 3 paths of length 2 to one of the two vertices; is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C ₃. In addition, we claim that for has the minimal energy among all the graphs considered while for has the minimal energy.      

Characterizations for Some Types of DNA Graphs

Vertex induced subgraphs of directed de Bruijn graphs with labels of fixed length k and over α letter alphabet are (α,k)-labelled. DNA graphs are (4,k)-labelled graphs. Pendavingh et al. proved that it is NP-hard to determine the smallest value α _k (D) for which a directed graph D can be (α _k (D),k)-labelled for any fixed . In this paper, we obtain the following formulas: and for cycle C _n and path P _n . Accordingly, we show that both cycles and paths are DNA graphs. Next we prove that rooted trees and self-adjoint digraphs admit a (Δ,k)-labelling for some positive integer k and they are DNA graphs if and only if Δ ≤ 4, where Δ is the maximum number in all out-degrees and in-degrees of such digraphs.     

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 , 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. The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined.     

On the path-Zagreb matrix

The definition of the path-Zagreb matrix for (chemical) trees PZ and its generalization to any (molecular) graph is presented. Additionally, the upper bound of , where G _n is a graph with n vertices is given.     

Minimal energy of unicyclic graphs of a given diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. For a given positive integer d with , we characterize the graphs with minimal energy in the class of unicyclic graphs with n vertices and a given diameter d.      

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Symmetrical features of the structure of fullerene C20

It is shown that fullerenes C₂₀ of the $ \bar 5\bar 3 It is shown that fullerenes C₂₀ of the m symmetry can form seven types of low-symmetry species of the initial “mother” fullerene molecule by means of continuous deformation. Possible routes of formation and continuous transformations of these species are presented on the adjacency graph of structural elements of fundamental regions of fullerenes C₂₀. The approach used in this work can be applied for design of structural-geometrical models of all low-symmetry species of fullerene C₂₀. Original Russian Text ? V.M. Talanov, N.V. Fedorova, V.V. Gusarov, 2009, published in Zhurnal Obshchei Khimii, 2009, Vol. 79, No. 2, pp. 285–288.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

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 be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

New Lower Bound on the Number of Perfect Matchings in Fullerene Graphs

A fullerene graph is a cubic and 3-connected plane graph (or spherical map) that has exactly 12 faces of size 5 and other faces of size 6, which can be regarded as the molecular graph of a fullerene. T. Doli [3] obtained that a fullerene graph with p vertices has at least (p+2)/2 perfect matchings by applying the recently developed decomposition techniques in matching theory of graphs. This note gets a better lower bound 3(p+2)/4 of the number of perfect matchings of a fullerene graph by finding its 2-extendability. This property further implies a chemical consequence that every derivative of a fullerene by substituting any two pairs of adjacent carbon atoms permits a Kekulé structure.     

New upper bounds on Zagreb indices

The first Zagreb index M ₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M ₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M ₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ₁), second maximum vertex degree (Δ₂) and minimum vertex degree (δ). Using this result we find an upper bound on M ₂(G). Moreover, we present upper bounds on and in terms of n,  m, Δ₁, Δ₂, δ, where denotes the complement of G.     

Unicycle graphs with extremal Merrifield–Simmons Index

The Merrifield–Simmons index f(G) of a (molecular) graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By we denote the set of unicycle graphs in which the length of its unique cycle is k. In this paper, we investigate the Merrifield–Simmons index f(G) for an unicycle graph G in . Unicycle graphs with the largest or smallest Merrifield–Simmons index are uniquely determined.     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

Distance-restricted matching extendability of fullerene graphs

A fullerene graph is a 3-connected cubic plane graph whose all faces are bounded by 5- or 6-cycles. In this paper, we show that a matching M of a fullerene graph can be extended to a perfect matching if the following hold: (i) three faces around each vertex in \(\{x,y:xy\in M\}\) are bounded by 6-cycles and (ii) the edges in M lie at distance at least 13 pairwise.     

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

On the Permanental Polynomials of Some Graphs

Let G be a simple graph with adjacency matrix A(G) and (G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Kleins idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results.1.If G is a bipartite graph containing no subgraph which is an even subdivision of K _2,3, then G has an orientation G ^e such that (G,x) = det (xI-A(G ^e )), where A(G ^e ) denotes the skew adjacency matrix of G ^e.2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph with 2n+2 vertices such that (G,x) is a factor of .3.Let T be an arbitrary tree with n vertices. Then , where ₁ , ₂ , ..., _n are the eigenvalues of T.     

On the Randić Index of Unicyclic Conjugated Molecules

The Randić index R(G) of a graph G is the sum of the weights of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.     

Unicycle Graphs with Maximum Generalized Topological Indices

Let G be an unicycle graph and d _v the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph , , where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized. This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan Province(02C210)(04B047).