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.     

Conjugated chemical trees with minimal energy and prescribed diameter

The energy of a molecular graph G is defined as the sum of the absolute values of the eigenvalues of A(G), where A(G) is the adjacency matrix of this graph. This article characterizes conjugated chemical trees with prescribed diameter and minimal energies and presents explicit expressions of their Hosoya indices. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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).     

On the Randić Index

The Randić index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of Delorme et al., Discrete Math. 257 (2002) 29), and then we will show that the result holds when δ(G)≥ 2.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion

The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z _G ) was discussed. Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director     

Distances and volumina for graphs

It has long been realized that connected graphs have some sort of geometric structure, in that there is a natural distance function (or metric), namely, the shortest-path distance function. In fact, there are several other natural yet intrinsic distance functions, including: the resistance distance, correspondent “square-rooted” distance functions, and a so‐called “quasi‐Euclidean” distance function. Some of these distance functions are introduced here, and some are noted not only to satisfy the usual triangle inequality but also other relations such as the “tetrahedron inequality”. Granted some (intrinsic) distance function, there are different consequent graph-invariants. Here attention is directed to a sequence of graph invariants which may be interpreted as: the sum of a power of the distances between pairs of vertices of G, the sum of a power of the “areas” between triples of vertices of G, the sum of a power of the “volumes” between quartets of vertices of G, etc. The Cayley–Menger formula for n-volumes in Euclidean space is taken as the defining relation for so-called “n-volumina” in terms of graph distances, and several theorems are here established for the volumina-sum invariants (when the mentioned power is 2). This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.     

Maximum energy trees with two maximum degree vertices

The energy E of a graph G is equal to the sum of the absolute values of the eigenvalues of G. In 2005 Lin et al. determined the trees with a given maximum vertex degree Δ and maximum E, that happen to be trees with a single vertex of degree Δ. We now offer a simple proof of this result and, in addition, characterize the maximum energy trees having two vertices of maximum degree Δ.     

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).     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

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.     

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).      

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.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

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 R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

On the Randić index

The Randić index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Discrete Mathematics 257, 29–38 by Delorme et al. gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of [2002]), and then we will show that the result holds when δ(G) ≥ 2.AMS subject classification: 05C18     

Kekulé structures of hexagonal chains—some unusual connections

It is known since 1977 that the number K of Kekulé structures of a hexagonal chain is equal to the topological Z-index of a pertinently constructed “caterpillar” tree. Based on this relation we now connect K with some of other, seemingly unrelated, concepts: continuants (from number theory) and matchings of the path–graph (further related to Fibonacci numbers). We also arrive at a tridiagonal determinantal expression for K.     

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 number of spanning trees in an (r,s)‐semiregular graph and its line graph

For a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.     

Two metrics for a graph-theoretical model of organic chemistry

A graph-theoretical model of organic chemistry is proposed. The main idea behind this model is a molecular graph in the form of a multigraph with loops; its vertices are evaluated by vertex labels (atomic symbols). The chemical distance between two graphs from the same family of isomeric graphs is based on the maximal common subgraph. The produced reaction graph is composed of the minimal number of edges and/or loops. The reaction distance assigned to the chemical transformationG ₁ G ₂ is equal to the minimal number of the so-called elementary transformations that are necessary for the transformation ofG ₁ intoG ₂ Because these metrics are not isometric, the resulting reaction graphs may depend on the metric used.Dedicated to the memory of Professor Milan Sekanina