Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ . . .≥ μn−1 > μn = 0. The Kirchhoff index of G is defined as Kf = Kf(G) = n∑n−1 k=1 1/μk. In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus–Gaddum-type result for the Kirch...

Several topological indices are known to have widespread implications in a variety of research areas. Over the years, Kirchhoff index has turned out be an extremely significant and efficient index. In this paper, we propose exact formulas for expected values random polyomino chain construct multiplicative degree-Kirchhoff additive We also carefully examine highest degree through

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a b s t r a c t In this paper, closed-form formulae for the Kirchhoff index and resistance distances of the Cayley graphs ov...

Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdős-Rényi random graphs. Since the graphs are rand...

Here we consider a class of generalized linear chains; that is, the ladder–like chains as a perturbation of a 2n–path by adding consecutive weighted edges between opposite vertices. This class of chains in particular includes a big family of networks that goes from the cycle, unicycle chains up to ladder networks. In this paper, we obtain the Green function, the effective resistance and the Kir...

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...