Spectra of graphs and applications 2016 (SGA): Belgrade, Serbia: May 18–20, 2016: book of abstracts
First page
35
Last page
36
Abstract
%The Kirchhoff Index was introduced as an alternative to other parameters used to discriminate among different molecules with similar shapes and structures. The Kirchhoff index is defined as the sum of the effective resistances of all the vertices of the (molecular) graph conventionally used to represent the topology of a chemical compound, where edge weights correspond to bond properties. It is known that the Kirchhoff index coincides with the sum of the inverses of the (non null) eigenvalues...
%The Kirchhoff Index was introduced as an alternative to other parameters used to discriminate among different molecules with similar shapes and structures. The Kirchhoff index is defined as the sum of the effective resistances of all the vertices of the (molecular) graph conventionally used to represent the topology of a chemical compound, where edge weights correspond to bond properties. It is known that the Kirchhoff index coincides with the sum of the inverses of the (non null) eigenvalues of the generalized inverse of the Laplacian matrix of the graph.
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%In this work we consider a class of generalized linear chains obtained from a $2n$--path, $P$, by the addition of weighted edges between appropriate vertices in order to model different chemical compounds. A generalized linear chain is seen as a perturbation of a path and we obtain its Kirchhoff index as non trivial functions of the corresponding expression for the path $P$. To this end, we deal with some $(s \times s)$--resistance matrix that holds all the information considered as the perturbation that makes a path to become a generalized chain. This approach requires the computation of the inverse of a tridiagonal $M$--matrix that leads to solve second order difference equations.