PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

The PI index of phenylenes

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

A Method of Computing the PI Index of Benzenoid Hydrocarbons Using Orthogonal Cuts

The Padmakar–Ivan (PI) index of a graph G is defined as PI , where for edge e=(u,v) are the number of edges of G lying closer to u than v, and is the number of edges of G lying closer to v than u and summation goes over all edges of G. The PI index is a Wiener–Szeged-like topological index developed very recently. In this paper, we describe a method of computing PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal cuts in a benzenoid system (H) and the edge number of H – a task significantly simpler than the calculation of PI index directly from its definition. On the eve of 70th anniversary of both Prof. Padmakar V. Khadikar and his wife Mrs. Kusum Khadikar.     

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.     

On bicriticality of (3,6)-fullerene graphs

A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least \(2n+2\) vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, \(G-u-v\) has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant.     

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.     

On the Randić Index of Quasi-tree Graphs

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v ₁,v ₂,...,v _n }. The 2-degree of v _i , denoted by t _i , is the sum of degrees of the vertices adjacent to . Let σ _i be the sum of the 2-degree of vertices adjacent to v _i . In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t _i , and σ _i , from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.     

Hosoya polynomials of TUC4C8(S) nanotubes

The Hosoya polynomial of a chemical graph G is , where d _G (u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived. This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809).     

Maximum Randić index on Trees with <Emphasis Type="Italic">k</Emphasis>-pendant Vertices

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T.     

The (n,n)-graphs with the first three extremal Wiener indices

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e., , where d _G (u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula.     

On the connectivity index of trees

The connectivity index χ₁(G) of a 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. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

Extremal hexagonal chains concerning k-matchings and k-independent sets

Denote by _n the set of the hexagonal chains with n hexagons. For any B _{n n} , let m _k (B _n ) and i _k (B _n ) be the numbers of k-matchings and k-independent sets of B _n , respectively. In the paper, we show that for any hexagonal chain B _{n n} and for any k0, m _k (L _n )m _k (B _n )m _k (Z _n ) and i _k (L _n )i _k (B _n )i _k (Z _n ), with left equalities holding for all k only if B _n =L _n , and the right equalities holding for all k only if B _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. These generalize some related results known before.     

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.     

Extremal k *-Cycle Resonant Hexagonal Chains

Denote by ^* _n the set of all k ^*-cycle resonant hexagonal chains with n hexagons. For any B _n ^* _n , let m(B _n ) and i(B _n ) be the numbers of matchings (=the Hosoya index) and the number of independent sets (=the Merrifield–Simmons index) of B _n , respectively. In this paper, we give a characterization of the k ^*-cycle resonant hexagonal chains, and show that for any B _n ^* _n , m(H _n )m(B _n ) and i(H _n )i(B _n ), where H _n is the helicene chain. Moreover, equalities hold only if B _n =H _n .     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

An exact expression for the wiener index of a TUC<Subscript>4</Subscript>C<Subscript>8</Subscript>(R) nanotorus

The Wiener index of a graph G is defined as , where V(G) is the set of all vertices of G and for denotes the length of a minimal path between x and y. A C ₄ C ₈ net is a trivalent decoration made by alternating squares C ₄ and octagons C ₈. It can cover either a cylinder or a torus. In this paper, an algorithm for computing the distance matrix of a C ₄ C ₈(R) nanotorus T = T[p,q] is given. Using this matrix, the Wiener index of T is computed.     

On the Randić Index of Acyclic Conjugated Molecules

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. We give a sharp lower bound on the Randić index of conjugated trees (trees with a perfect matching) in terms of the number of vertices. A sharp lower bound on the Randić index of trees with a given size of matching is also given Mei Lu: Partially supported by NNSFC (No. 60172005) Lian-zhu Zhang: Partially supported by NNSFC (No. 10271105) Feng Tian: Partially supported by NNSFC (No. 10431020)