Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks

Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and the spectral gap of the transition probability matrix associated to the random walk on the graph. We then use bounds of the spectral gap of more specialized graphs, available in the literature, in order to obtain upper bounds for the Kirchhoff index of these specialized graphs. As a byproduct, we obtain a closed‐form formula for the Kirchhoff index of the d‐dimensional cube in terms of the first inverse moment of a positive binomial variable. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

A formula for the Kirchhoff index

We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed‐form formula for the effective resistance between any pair of vertices when the considered network has some symmetries, which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n‐th formula.     

New Nordhaus-Gaddum-type results for the Kirchhoff index

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstić (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.     

Bounds for the Randic connectivity index

For a saturated hydrocarbon with n carbon atoms and m carbon-carbon bonds and with Randic connectivity index chi, two functions, L = L(n,m) and U = U(n,m), are determined, such that L < or = chi < U. These bounds are better than those previously reported; for most chemically relevant values of n and m there exist hydrocarbons for which chi = L or chi = U.     

Closed‐form formulas for Kirchhoff index

We find closed‐form expressions for the resistance, or Kirchhoff index, of certain connected graphs using Foster's theorems, random walks, and the superposition principle. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 135–140, 2001     

On the Kirchhoff index of regular graphs

Using probabilistic tools, we give a compact formula for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and Kemeny's constant, as well as general upper and lower bounds in terms of d and N. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

On resistance-distance and Kirchhoff index

We provide some properties of the resistance-distance and the Kirchhoff index of a connected (molecular) graph, especially those related to its normalized Laplacian eigenvalues.     

Kirchhoff index of linear hexagonal chains

The resistance distance r_ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L_n consists of the Laplacian spectrum of path P_2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L_n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L_n is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including L_n. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Kirchhoff index of linear pentagonal chains

The resistance distance r_ij between two vertices v_i and v_j of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed‐form formulae of the Kirchhoff index of liner pentagonal chain P_n in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of P_n is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Bounds on Harary index

In this paper, we obtain the lower and upper bounds on the Harary index of a connected graph (molecular graph), and, in particular, of a triangle- and quadrangle-free graphs in terms of the number of vertices, the number of edges and the diameter. We give the Nordhaus–Gaddum-type result for Harary index using the diameters of the graph and its complement. Moreover, we compare Harary index and reciprocal complementary Wiener number for graphs.     

Deletion and contraction identities for the resistance values and the Kirchhoff index

We establish identities, which we call deletion and contraction identities, for the resistance values on an electrical network. As an application of these identities, we give an upper bound to the Kirchhoff index of a molecular graph. Our upper bound, expressed in terms of the set of vertices and the edge connectivity of the graph, improves previously known upper bounds. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Another look at the degree‐Kirchhoff index

Let G be an arbitrary graph with vertex set {1,2, …,N} and degrees d_i ≤ D, for fixed D and all i, then for the index R′(G) = ∑_{i < j}d_id_jR_ij we show that We also show that the minimum of R′(G) over all N‐vertex graphs is attained for the star graph and its value is 2N² ? 5N + 3. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Preconcentration techniques for trace analysis via neutron activation

A review is given of the methods that have been proposed for enrichment of trace elements in samples that are to be analysed by neutron-activation methods. The emphasis is on classification of methods, with full illustrations by means of practical examples.     

Bounds for reactivity indices

Lower and upper bounds are derived for bond number, localization energy and atom self-polarizability of alternant hydrocarbons. It is proved that in acyclic polyenes the maximal bond number is 1, 2 and 3, respectively for primary, secondary and tertiary carbon atoms.     

Atomic term symbols via partitioning techniques

Two techniques for obtaining Russell–Saunders term symbols (terms) for equivalent electrons is presented. Both techniques are based on partitioning the n equivalent electrons into a set of α electrons and a set of β electrons. Fundamental tables are presented that give the number of states for the α or β electrons for all possible values M(M) and S^α(S^β). One interpretation of these tables leads to the total number of states of the n equivalent electrons for values of L and S. From this the number of terms for L and S is easily determined. Another interpretation of the fundamental tables gives the maximum number of terms for n equivalent electrons with values of L and S. From this the actual number of terms is obtained. Examples of equivalent p, d, f, and g electrons are provided. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011