On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

On relation between the Kirchhoff index and number of spanning trees of graph

Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,be a simple connected graph, with sequence of vertex degrees$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning tree...

note on degree kirchhoff index of graphs

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

Computing the additive degree-Kirchhoff index with the Laplacian matrix

For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.

8.1 Min Degree Spanning Tree

This problem cab be shown to be NP-Hard by reducing Hamiltonian path to it. Essentially existence of a spanning tree of max degree two is equivalent to a having a Hamiltonian path in the graph. This also shows that the best approximation we can hope for (unless P = NP) is ∆∗ + 1 where ∆∗ is the max degree of the optimal tree. Note that we usually express approximations in a multiplicative form,...

An Interlacing Result on Normalized Laplacians