The Kirchhoff Index of Toroidal Meshes and Variant Networks

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices in G. In this paper, we established the relationships between the toroidal meshes network Tm×n and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of L(Tm×n), S(Tm×n), T(Tm×n), and C(Tm×n) were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach

Effective resistances and Kirchhoff index in subdivision networks

We define a subdivision network ¿S of a given network ¿; by inserting a new vertex in every edge, so that each edge is replaced by two new edges with conductances that fulfill electrical conditions on the new network. In this work, we firstly obtain an expression for the Green kernel of the subdivision network in terms of the Green kernel of the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a wheel.Peer ReviewedPostprint (author's final draft

The Kirchhoff Index of Hypercubes and Related Complex Networks

The resistance distance between any two vertices of G is defined as the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices in G. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networks Qn by utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networks Qn and its three variant networks l(Qn), s(Qn), t(Qn) by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes of l(Qn), s(Qn), and t(Qn) were proposed, respectively

Green’s kernel of Schrödinger operators on generalized subdivision networks

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Green's function in partial subdivision networks

In the present work, we define a partial subdivision network of a given network G, by inserting a new vertex in some selected edges of G, so that each of these edges is replaced by two new edges with conductances that fulfil the Kirchhoff series law on the new network. Then, we obtain an expression for the Green kernel of the partial subdivision network in terms of the Green kernel of the base network. For that, we show the relation between Poisson problems on the partial subdivision network and Poisson problems on the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the partial subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a star network in which we have subdivided the even edges.Peer ReviewedPostprint (author's final draft