The degree Kirchhoff index of weighted graphs. (English) Zbl 1499.05137
For a weighted graph given the resistance \(\omega(i,j)\) on the edge \(ij\), then the conductance is defined as \(c_{ij}=1/\omega(i,j)\) (\(c_{ij}=0\) if vertices \(i\) and \(j\) are nonadjacent). The weighted degree Kirchhoff index of \(G\) is defined as \(Kf^\prime(G)=\sum_{i<j}c_{i}c_{j}R(i,j)\), where \(R(i,j)\) is the resistance distance between the vertices \(i\) and \(j\). The aim of this paper is to establish several lower bounds for degree Kirchhoff index of weighted graphs, by extending some lower bounds deduced by J. L. Palacios and J. M. Renom [“Another look at the degree Kirchhoff index”, Int. J. Quantum Chem. 111, No. 14, 3453–3455 (2011; doi:10.1002/qua.22725)]. The following result plays a crucial role in this paper: \(R(i,j)\geq 1/c_{i}+1/c_{j}\) if \(ij\notin E(G)\) and \(R(i,j)\geq(c_i+c_j-2c_{ij})/(c_ic_j-c_{ij}^{2})\) otherwise, where \(c_i=\sum_{k\in N(i)}c_{ik}\) and \(N(i)\) is the set of neighbors of vertex \(i\).
Reviewer: Ioan Tomescu (Bucureşti)
MSC:
05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
05C12 | Distance in graphs |
05C22 | Signed and weighted graphs |