We consider a canonical ensemble of synchronization-optimized networks of identical oscillators under external noise. By performing a Markov chain Monte Carlo simulation using the Kirchhoff index, i.e., the sum of the inverse eigenvalues of the Laplacian matrix (as a graph Hamiltonian of the network), we construct more than 1,000 different synchronization-optimized networks. We then show that t...

Let G be a connected graph. The resistance distance between any two vertices of is equal to the effective them in corresponding electrical network constructed from by replacing each edge with unit resistor. Kirchhoff index defined as sum distances all pairs vertices. Recently, Yang and Wang determined maximum phenylene chains proposed conjecture about minimum index. In this note, we characteriz...

In this paper, we determine the Randić energy of m-splitting graph, m-shadow graph and m-duplicate a given m being an arbitrary integer. Our results allow construction infinite sequence graphs having same energy. Further, some invariants like degree Kirchhoff index, Kemeny’s constant number spanning trees special graphs. From our results, indicate how to obtain infinitely many pairs equienerget...

In 1993 two novel distance-based topological indices were put forward. In the case of acyclic molecular graphs both are equal to the Wiener index, but both differ from it if the graphs contain cycles. One index is defined (Mohar, B.; Babić, D.; Trinajstić, N. J. Chem. Inf. Comput. Sci. 1993, 33, 153-154) in terms of eigenvalues of the Laplacian matrix, whereas the other is conceived (Klein, D. ...