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 largest eigenvalues of trees with perfect matchings

Let λ₁ (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let be the set of trees with perfect matchings on 2m vertices, and . Among the trees in , we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when . Furthermore, it is proved that, for two trees T ₁ and T ₂ in (m≥ 4), if and Δ (T ₁) > Δ (T ₂), then λ₁ (T ₁) > λ₁ (T ₂).     

On Acyclic Molecular Graphs with Prescribed Numbers of Edges that Connect Vertices with given Degrees

We find a necessary and sufficient conditions on a sequence

Extremal (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">n</Emphasis> + 1)-graphs with respected to zeroth- order general Randić index

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. If d _v denotes the degree of the vertex v, then the zeroth-order general Randić index of the graph G is defined as , where α is a real number. We characterize, for any α, the (n,n + 1)-graphs with the smallest and greatest zeroth-order general Randić index.     

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).      

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.     

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).     

On the spectral radius of graphs with connectivity at most <Emphasis Type="Italic">k</Emphasis>

In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k. We show that among those graphs, the maximal spectral radius is obtained uniquely at , which is the graph obtained by joining k edges from k vertices of K _n-1 to an isolated vertex. We also show that the spectral radius of will be very close to n − 2 for a fixed k and a sufficiently large n.     

Sharp upper bounds for total π-electron energy of alternant hydrocarbons

The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V ₁, V ₂; E) is a bipartite graph with edges and , then

The first and second largest Merrifield–Simmons indices of trees with prescribed pendent vertices

The Merrifield–Simmons index of a 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 T(n,k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield–Simmons index for a tree T in T(n,k). For all trees in T(n,k), we determined unique trees with the first and second largest Merrifield–Simmons index, respectively.     

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.     

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 derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives and , all with m > n, are also evaluated.     

On the Laplacian spectral radii of trees with perfect matchings

Denote by the set of trees of order 2k with perfect matchings. GUO [Guo, Linear Algebra Appl. 368:379–385, 2003.] determined the largest value of Laplacian spectral radii μ(T) of the trees T in and gave the corresponding tree T in whose μ(T) reaches this largest value. In this paper, we determine the second to the sixth largest values of μ(T) of the trees T in and also give the corresponding trees T in whose μ(T) reach these values.     

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     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

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.     

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.      

On the special values of monic polynomials of hypergeometric type

Special values of monic polynomials y _n (s), with leading coefficients of unity, satisfying the equation of hypergeometric type