724 research outputs found
Kirchhoff index of composite graphs
AbstractLet G1+G2, G1∘G2 and G1{G2} be the join, corona and cluster of graphs G1 and G2, respectively. In this paper, Kirchhoff index formulae of these composite graphs are given
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 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
Cascading Failures in Power Grids - Analysis and Algorithms
This paper focuses on cascading line failures in the transmission system of
the power grid. Recent large-scale power outages demonstrated the limitations
of percolation- and epid- emic-based tools in modeling cascades. Hence, we
study cascades by using computational tools and a linearized power flow model.
We first obtain results regarding the Moore-Penrose pseudo-inverse of the power
grid admittance matrix. Based on these results, we study the impact of a single
line failure on the flows on other lines. We also illustrate via simulation the
impact of the distance and resistance distance on the flow increase following a
failure, and discuss the difference from the epidemic models. We then study the
cascade properties, considering metrics such as the distance between failures
and the fraction of demand (load) satisfied after the cascade (yield). We use
the pseudo-inverse of admittance matrix to develop an efficient algorithm to
identify the cascading failure evolution, which can be a building block for
cascade mitigation. Finally, we show that finding the set of lines whose
removal has the most significant impact (under various metrics) is NP-Hard and
introduce a simple heuristic for the minimum yield problem. Overall, the
results demonstrate that using the resistance distance and the pseudo-inverse
of admittance matrix provides important insights and can support the
development of efficient algorithms
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
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